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CHAPTER 6
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Existence and Stability of Spikes
for the Gierer–Meinhardt System
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Juncheng Wei
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Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
E-mail: wei@math.cuhk.edu.hk
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Contents
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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Steady states in shadow system case . . . . . . . . . . . . . . . . . . . .
2.1. Reduction to single equation . . . . . . . . . . . . . . . . . . . . .
2.2. Subcritical case: spikes to (2.4) . . . . . . . . . . . . . . . . . . .
2.3. Localized energy method (LEM) . . . . . . . . . . . . . . . . . . .
2.4. Bubbles to (2.4): the critical case . . . . . . . . . . . . . . . . . . .
2.5. Bubbles to (2.4): slightly supercritical case . . . . . . . . . . . . .
2.6. Concentration on higher-dimensional sets . . . . . . . . . . . . . .
2.7. Robin boundary condition . . . . . . . . . . . . . . . . . . . . . .
3. Stability and instability in the shadow system case . . . . . . . . . . . .
3.1. Small eigenvalues for L . . . . . . . . . . . . . . . . . . . . . . .
3.2. A reduction lemma . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Large eigenvalues: NLEP method . . . . . . . . . . . . . . . . . .
3.4. Uniqueness of Hopf bifurcations . . . . . . . . . . . . . . . . . . .
3.5. Finite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6. The stability of boundary spikes for the Robin boundary condition
4. Full Gierer–Meinhardt system: One-dimensional case . . . . . . . . . .
4.1. Bound states: the case of Ω = R1 . . . . . . . . . . . . . . . . . .
4.2. The bounded domain case: Existence of symmetric K-spikes . . .
4.3. The bounded domain case: existence of asymmetric K-spikes . . .
4.4. Classification of asymmetric patterns . . . . . . . . . . . . . . . .
4.5. The stability of symmetric and asymmetric K-spikes . . . . . . .
5. The full Gierer–Meinhardt system: Two-dimensional case . . . . . . .
5.1. Bound states: spikes on polygons . . . . . . . . . . . . . . . . . .
5.2. Existence of symmetric K-spots . . . . . . . . . . . . . . . . . . .
5.3. Existence of multiple asymmetric spots . . . . . . . . . . . . . . .
5.4. Stability of symmetric K-spots . . . . . . . . . . . . . . . . . . . .
5.5. Stability of asymmetric K-spots . . . . . . . . . . . . . . . . . . .
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HANDBOOK OF DIFFERENTIAL EQUATIONS
Stationary Partial Differential Equations, volume 5
Edited by M. Chipot
© 2008 Elsevier B.V. All rights reserved
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6. High-dimensional case: N 3
7. Conclusions and remarks . . .
Acknowledgments . . . . . . . .
References . . . . . . . . . . . .
F:chipot506.tex; VTEX/Rita p. 2
J. Wei
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576
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Existence and stability of spikes
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1. Introduction
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It is a common belief that diffusion is a smoothing and trivializing process. Indeed, this is
the case for a single diffusion equation. Consider the heat equation
⎧
⎨ ut = u
u(x, 0) = u0 (x) 0
⎩ ∂u
∂ν = 0
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(1.1)
in Ω × (0, +∞),
in Ω,
on ∂Ω × (0, +∞),
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(1.2)
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⎧ u = D u + f (u, v)
t
u
⎪
⎪
⎨ v = D v + g(u, v)
t
v
u(x, 0) = u (x),
⎪
⎪
⎩ ∂u ∂v 0
∂ν = ∂ν = 0
v(x, 0) = v0 (x)
in Ω × (0, +∞),
in Ω × (0, +∞),
in Ω,
on ∂Ω × (0, +∞).
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(1.3)
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In 1957, Turing [68] proposed a mathematical model for morphogenesis, which describes the development of complex organisms from a single shell. He speculated that localized peaks in concentration of a chemical substance, known as an inducer or morphogen,
could be responsible for a group of cells developing differently from the surrounding cells.
He then demonstrated, with linear analysis, how a non-linear reaction diffusion system
like (1.3) could possibly generate such isolated peaks. Later in 1972, Gierer and Meinhardt
[21] demonstrated the existence of such solution numerically for the following (so-called
Gierer–Meinhardt system)
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it is known that when Ω is convex, the only stable solutions are constants [5,46]. Thus
there are only trivial patterns (constant solutions) for single reaction–diffusion equations
(on convex domains).
On the other hand, it is important to be able to use diffusion (and reaction) to model
pattern formations in various branches of science (e.g., biology and chemistry). One important question is: can we get non-trivial patterns (stable non-trivial solutions) for systems
of reaction–diffusion equations?
Let us consider the following system of reaction–diffusion equations:
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in Ω × (0, +∞),
in Ω,
on ∂Ω × (0, +∞).
Assume that u0 (x) iscontinuous. It is known that u(x, t) is smooth for t > 0 (smooth1
ing), and u(x, t) → |Ω|
Ω u0 (x) dx as t → +∞ (trivializing). A similar result holds when
a source/sink term (or a reaction term) is present. Namely, for the problem
⎧
⎨ ut = u + f (u)
u(x, 0) = u0 (x) 0
⎩ ∂u
∂ν = 0
3
(GM)
⎧ ∂a
ap
2
⎪
⎨ ∂t = a − a + hq ,
r
τ ∂h
= Dh − h + ahs ,
∂t
⎪
⎩ ∂a ∂h
∂ν = ∂ν = 0,
x ∈ Ω, t > 0,
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x ∈ Ω, t > 0,
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x ∈ ∂Ω.
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J. Wei
Here, the unknowns a = a(x, t) and h = h(x, t) represent the respective concentrations at
point x ∈ Ω ⊂ RN and at time t of the biochemical called an activator and an inhibitor;
∂2
> 0, D > 0, τ > 0 are all positive constants; = N
2 is the Laplace operator in
j =1
∂xj
RN ; Ω is a smooth bounded domain in RN ; ν(x) is the unit outer normal at x ∈ ∂Ω. The
exponents (p, q, r, s) are assumed to satisfy the condition
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p > 1,
q > 0,
r > 0,
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s 0,
and γ :=
at = −a + a p / hq ,
τ ht = −h + a r / hs .
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qr
> 1.
(p − 1)(s + 1)
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Gierer–Meinhardt system was used in [21] to model head formation in the hydra. Hydra, an animal of a few millimeters in length, is made up of approximately 100,000 cells
of about fifteen different types. It consists of a “head” region located at one end along
its length. Typical experiments on hydra involve removing part of the “head” region and
transplanting it to other parts of the body column. Then, a new “head” will form if and only
if the transplanted area is sufficiently far from the (old) head. These observations have led
to the assumption of the existence of two chemical substances—a slowly diffusing (i.e.,
1) activator a and a fast diffusing (i.e., D ) inhibitor h.
To understand the dynamics of (GM), it is helpful to consider first its corresponding
“kinetic system”
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(1.4)
qr
it
This system has a unique constant steady state a ≡ 1, h ≡ 1. For 0 < τ < (p−1)(s+1)
is easy to see that the constant solution a ≡ 1, h ≡ 1 is stable as a steady state of (ODE).
However, if √ is small, it is not hard to see that the constant steady state a ≡ 1, h ≡
D
1 of (GM) becomes unstable and bifurcation may occur. This phenomenon is generally
referred to as Turing’s diffusion-driven instability. (A general criteria for this can be found
in Murray’s book [47].)
There are many other reaction–diffusion systems which exhibit Turing’s diffusiondriven instability: they include Gray–Scott model from chemical reactor theory, Schnakenberg model, Sel’kov model, Lengyl–Epstein model, Thomas model, Keener–Tyson model,
Brusselator, Oregonator, etc. For introduction and discussion on these general Turing models, we refer to the book [47]. A survey of mathematical modeling of biological and chemical phenomena using reaction–diffusion systems is given in [38]. Mathematical modeling
of patterns in biological morphogenesis using extensions of GM model are discussed in
[36] and [48].
Several common characteristics of Turing type reaction–diffusion systems include: first,
they are non-variational, i.e., they do not have Lyapunov or energy functional so standard
variational (or energy) method cannot be applied; second, they are non-cooperative, i.e.,
they do not have Maximum Principles so sub-super-solution method cannot be applied;
third, they support finite-amplitude spatial-temporal patterns of remarkable diversity and
complexity, such as stable spikes, layers, stripes, spot-splitting, traveling waves, etc. (See
[63].) The study of these RD systems not only increases our knowledge on Turing patterns,
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Existence and stability of spikes
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but also induces new tools and techniques to deal with other problems which may share
similar characteristics.
The most interesting phenomena associated with (GM) is the existence of stable spikes
and stripes. The numerical studies of [21] and more recent those of [31] have revealed
that in the limit → 0, the (GM) system seems to have stable stationary solutions with
the property that the activator concentration is localized around a finite number of points
in Ω̄. Moreover, as → 0, the pattern exhibits a “spike layer phenomenon” by which we
mean that the activator concentration is localized in narrower and narrower regions around
some points and eventually shrinks to a certain number of points as → 0, whereas the
maximum value of the activator concentration diverges to +∞.
Such kind of point-condensation phenomena has generated a lot of interests both mathematically and biologically in recent years. The purpose of this paper is to report on the
current trend and status of such studies (up to June, 2006). We shall not give most of proofs.
For more details, please see the references and therein.
In the study of spiky patterns (or concentration phenomena), two fundamental methods
emerge. The first one is the so-called “Localized Energy Method”, or LEM in short. LEM
is a combination of traditional Lyapunov–Schmidt reduction method with variational techniques. This is a very useful tool to construct solutions with various concentration behavior,
such as spikes, layers, or vortices. The second method is the so-called “Nonlocal Eigenvalue Problem Method”, or NLEP in short. This deals with eigenvalue problems which
are non-selfadjoint. It plays fundamental role in the study of stability of spike patterns. In
this survey, I shall illustrate these two methods in details in the hope that they may find
applications in other problems.
Throughout this paper, unless otherwise stated, we always assume that
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D is finite,
τ 0.
(1.5)
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2.1. Reduction to single equation
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In general, the full (GM) system is very difficult to study. A very useful idea, which goes
back to Keener and Nishiura, is to consider the so-called shadow system. Namely, we let
D → +∞ first. Suppose that the quantity −h + a p / hq remains bounded, then we obtain
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2. Steady states in shadow system case
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h → 0,
∂h
= 0 on ∂Ω.
∂ν
(2.1)
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Thus h(x, t) → ξ(t), a constant. To derive the equation for ξ(t), we integrate both sides
of the equation for h over Ω and then we obtain the following so-called shadow system
⎧
2
p q in Ω,
⎪
⎨ at = a − a + a /ξ
1
r
s
τ ξt = −ξ + |Ω|
Ω a dx/ξ ,
⎪
⎩
a > 0 in Ω and ∂a
∂ν = 0 on ∂Ω.
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(2.2)
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J. Wei
The advantage of shadow system is that by a simple scaling,
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q
a = ξ p−1 u,
ξ=
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ur
2
p−1
(p−1)(s+1)−qr
3
,
(2.3)
Ω
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= 0 in Ω,
and ∂u
∂ν = 0 on ∂Ω
u > 0 in Ω
(2.4)
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whose energy functional is given by
J [u] :=
Ω
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1
1
p+1
dx,
|∇u|2 + u2 −
u
2
2
p+1 +
where u+ = max(u, 0),
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(2.5)
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2.2. Subcritical case: spikes to (2.4)
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+2
N +2
Let us assume first that 1 < p < ( N
N −2 )+ (= N −2 if N 3; = +∞ when N = 1, 2).
In this case, problem (2.4) can be studied by traditional variational methods, for example,
Mountain-Pass method, or Nehari’s solution manifold method. For Mountain-Pass method,
by taking a function e(x) ≡ k for some constant k in Ω, and choosing k large enough, we
have J (e) < 0, for all ∈ (0, 1). Then for each ∈ (0, 1), we can define the so-called
mountain-pass value
c = inf max J h(t)
h∈Γ 0t1
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for u ∈ H 1 (Ω).
First we give some definitions on solutions to (2.4). A family of solutions {u } to (2.4)
are called concentrated solutions if there exists a subset Γ ⊂ Ω̄ such that u → 0 in
0 (Ω̄\Γ ) and max
Cloc
x∈Γ u (x) c0 > 0. If Γ consists of only points in Ω̄, these kind
solutions are called point condensations. Among point condensations, there are two kinds:
spikes and bubbles. Spikes are those concentrated solutions such that maxx∈Ω̄ u C,
while bubbles are those with maxx∈Ω̄ u → +∞. If the dimension of Γ is positive, concentrated solutions are also called layers. (Similar definitions can also be given for solutions
of the full Gierer–Meinhardt system by considering the activator a only.)
In the following, we discuss the existence of all kinds of concentrated solutions to (2.4).
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2 u − u + up
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|Ω|
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the stationary shadow system can be reduced to a single equation
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inf
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38
(2.6)
39
40
41
where Γ = {h: [0, 1] → H 1 (Ω) | |h(t) is continuous, h(0) = 0, h(1) = e}.
It is easy to see that (Lemma 2.1 of [57]), c can be characterized by
c =
31
sup J [tu],
u≡0, u∈H 1 (Ω) t>0
42
43
(2.7)
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Existence and stability of spikes
1
2
3
4
5
493
which can be shown to be the least among all non-zero critical values of J . (This formulation (2.7) is sometimes referred to as the Nehari manifold technique.) Moreover, c is
attained by some function u which is then called a least-energy solution.
In a series of papers [57] and [58], Ni and Takagi studied the so-called least energy
solutions and proved the following theorem
6
7
8
9
17
18
= 0,
w(0) = maxy∈RN w(y),
w − w + w p
22
w > 0 in
w → 0 at ∞.
(2.8)
c = N
27
30
31
32
33
34
35
36
37
1
I [w] − c1 H (P ) + o()
2
40
41
44
45
15
17
20
21
22
25
(2.9)
26
27
28
where
I [w] =
RN
1
1
|∇w|2 + w 2 −
w p+1
2
p+1
29
30
31
32
2
order is given by
is the energy of the ground state. A further expansion of c up to the
[90]
2
2
N 1
2
I [w] − c1 H (P ) + c2 H (P ) + c3 R(P ) + o (2.10)
c = 2
33
34
35
36
37
38
where c1 , c2 , c3 are generic constants and R(P ) is the scalar curvature at P . In particular
c1 , c3 > 0. (When N = 2, a further expansion to the order of 3 is also given in [91].) Some
applications of the formula (2.10) are given in [90].
42
43
14
24
38
39
13
23
R EMARK 2.2.2. The proof of Theorem 2.1 is by expansion of energy:
26
29
9
19
R EMARK 2.2.1. The existence of ground state to (2.8) is well known. The radial symmetry
of w follows from the famous Gidas–Ni–Nirenberg theorem [22]. The uniqueness of w is
proved in [39].
25
28
8
18
23
24
7
16
RN ,
19
21
5
12
where H (P ) is the mean curvature function for P ∈ ∂Ω, and u (P + y) → w(y) uniformly in Ω,P = {y | P + y ∈ Ω}, where w(y) is the unique solution of the following
16
20
4
11
P ∈∂Ω
12
15
3
10
H (P ) → max H (P ),
11
14
2
6
T HEOREM 2.1. (See [57,58].) For sufficiently small, there exists a mountain-pass solution u which is also least-energy solution such that u has only one local maximum point
2 (Ω̄\{P }). Moreover, as → 0,
P ∈ ∂Ω and u → 0 in Cloc
10
13
1
39
40
41
42
Since then there has been a lot of studies on problem (2.4). A general principle is
that boundary spike solutions are related to the boundary mean-curvature H (P ), P ∈ ∂Ω,
while interior spike solutions are related to the distance function d(P , ∂Ω). Note also that
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J. Wei
d
for boundary spike the order is usually O() while for interior spikes the order is O(e− )
for some d > 0.
Let me mention some results on multiple boundary and interior peaked solutions.
For single and multiple boundary spikes, Gui [26] first constructed multiple boundary
spike solutions at multiple local maximum points of H (P ), using variational method. Wei
[73], Wei and Winter [82,83] (independently by Bates, Dancer and Shi [4]) constructed
single and multiple boundary spike solutions at multiple non-degenerate critical points of
H (P ), using Lyapunov–Schmidt reduction method. Y.Y. Li [41], del Pino, Felmer and
Wei [16] constructed single and multiple boundary spikes in the degeneracy case. Using
Localized Energy method (LEM), a clustered solution is also constructed by Gui, Wei and
Winter [29] (independently by Dancer and Yan [9]).
12
13
min H (P ) > min H (P ).
(2.11)
Γ
∂Γ
17
24
J [u ] = N
25
26
K
I [w] − c1 2
29
j =1
37
38
45
16
18
19
20
27
.
(2.12)
28
29
For single and multiple interior peaked solutions, the situation is quite different, as the
errors are exponentially small. Wei [79,74] first constructed single interior peak solution at
a strictly local maximum point of d(P , ∂Ω). Gui and Wei [27] proved the following
31
32
33
34
T HEOREM 2.3. (See [27].) For any fixed positive integer k, there exists k such that for
< k , problem (2.4) has a solution u with k interior local maximum points Pj, ∈ Ω.
Moreover, (P1, , . . . , Pk, ) approaches a limiting sphere-packing position, i.e.,
ϕk (P1, , . . . , Pk, ) →
max
(P1 ,...,Pk )∈Ω k
ϕk (P1 , . . . , Pk )
(2.13)
41
44
15
30
40
43
11
26
i=j
39
42
10
25
34
36
9
24
30
35
8
22
H (Pj, )
|Pi, − Pj, |
γ0 + o(1) w
−
28
33
7
23
K
27
32
6
21
The energy expansion for K-boundary spikes is
23
31
5
17
Then for any fixed positive integer k, there exists k such that for < k , problem (2.4) has
a solution u with k boundary local maximum points Pj, ∈ Γ . Furthermore, H (Pj, ) →
minΓ H (P ).
21
22
4
14
16
20
3
13
T HEOREM 2.2. (See [9,29].) Let Γ be a subset of ∂Ω, where it holds
15
19
2
12
14
18
1
35
36
37
38
39
40
41
42
where
ϕk (P1 , . . . , Pk ) = min |Pi − Pj |, 2d(Pl , ∂Ω) .
i,j,l,i=j
43
(2.14)
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Existence and stability of spikes
1
2
3
4
495
The energy expansion for K-interior spikes is
1
2
K
2d(Pj, ,∂Ω)
N
e−
J [u ] = KI [w] − γ0
3
4
j =1
5
|Pi, − Pj, | .
− γ1
w
6
7
5
6
(2.15)
i=j
8
9
10
11
12
13
16
17
20
23
26
27
28
29
30
31
1K αN,Ω,f
N (| ln |)N
where αN,Ω,p is a constant depending on N, Ω and p only, there exists a solution with K
interior peaks. (An explicit formula for αN,Ω,p is also given.) As a consequence, we obtain
α
] number of solutions. Moreover,
that for sufficiently small, there exists at least [ N (|N,Ω,p
ln |)N
for each β ∈ (0, N ) there exists solution with energy in the order of N −β .
32
33
34
12
13
15
16
17
19
20
21
T HEOREM 2.5. (See [44].) There exists an 0 > 0 such that for 0 < < 0 and for each
integer K bounded by
24
25
11
18
Theorems 2.2, 2.3 and 2.4 imply that the number of solutions to (2.4) goes to infinity as
→ 0. Recently, the following lower bound on number of solutions is obtained:
21
22
10
14
T HEOREM 2.4. (See [28].) For any two fixed positive integers k, l, there exists k,l such
that for < k,l , problem (2.4) has a solution u with k interior local maximum points and
l boundary maximum points.
18
19
8
9
Grossi, Pistoia and Wei [30] further showed that there is an one-to-one correspondence
between the (sub-differential) critical points of ϕk and k-interior peaked solutions.
Concerning the existence of mixed-boundary-interior-spikes, the following theorem
gives a complete answer.
14
15
7
22
23
24
25
26
27
28
29
30
31
32
Theorems 2.2, 2.3, 2.4 and 2.5 can all be proved by the powerful method—Localized
Energy Method—which was first introduced in [27]. We shall discuss it next.
33
34
35
35
36
36
37
2.3. Localized energy method (LEM)
38
39
40
41
42
43
44
45
37
38
We illustrate a general method in finding solutions with concentrating behavior—the socalled Localized Energy Method, or LEM in short. The advantage of such method is that
it can be applied to subcritical, critical or supercritical problems, as long as the limiting
solution is well analyzed. This method was introduced in Gui and Wei [27] in dealing with
spikes.
In the following, we show how to prove Theorem 2.5 by LEM. We need to introduce
some notation first.
39
40
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J. Wei
Theorem 2.5 actually holds for a slightly more general equation than (2.4), namely,
⎧
2
⎪
⎨ u − u + f (u) = 0
u>0
⎪
⎩ ∂u = 0
∂ν
2
3
in Ω,
in Ω,
on ∂Ω.
(2.16)
14
6
17
18
19
20
21
22
23
24
25
26
w > 0 in RN ,
w → 0 at ∞,
(2.17)
29
30
31
∂w
∂w
Kernel − 1 + f (w) = span
.
,...,
∂y1
∂yN
40
+2
One typical example of f is: f (u) = up − auq , where a 0, 1 < q < p < ( N
N −2 )+ . For
the uniqueness of w, see [39] and [40]. The proof of non-degeneracy is given in [58].
Without loss of generality, we may assume that 0 ∈ Ω. By the following rescaling:
x = z,
(2.19)
45
19
21
22
23
24
26
27
equation (2.16) becomes
28
29
u − u + f (u) = 0 in Ω ,
u > 0 in Ω , and ∂u
∂ν = 0 in ∂Ω .
(2.20)
30
31
32
For u ∈ H 2 (Ω ), we put
33
34
S [u] = u − u + f (u).
(2.21)
35
36
37
Then (2.20) is equivalent to
38
S [u] = 0,
u ∈ H 2 (Ω ),
u>0
in Ω ,
∂u
= 0 on ∂Ω .
∂ν
39
(2.22)
1
J˜ [u] =
2
Ω
|∇u|2 + u2 −
42
43
F (u),
Ω
40
41
Associated with problem (2.20) is the following energy functional
43
44
14
25
z ∈ Ω := {z | |z ∈ Ω},
41
42
13
20
38
39
11
18
(2.18)
36
37
10
17
34
35
9
16
has a unique solution w(y) and w is non-degenerate, i.e.,
32
33
8
15
27
28
7
12
w − w + f (w) = 0,
w(0) = maxy∈RN w(y),
15
16
4
5
We will always assume that f : R → R is of class C 1+σ for some 0 < σ 1 and satisfies
the following conditions (f1)–(f2):
(f1) f (u) ≡ 0 for u 0, f (0) = f (0) = 0.
(f2) The following equation
13
1
u ∈ H 1 (Ω ).
(2.23)
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Existence and stability of spikes
1
2
3
4
We define two inner products:
u, v =
uv, for u, v ∈ L2 (Ω );
7
8
9
12
13
14
15
2
(2.24)
(u, v) =
5
(∇u∇v + uv),
for u, v ∈ H (Ω ).
1
(2.25)
8
Let σ be the Hölder exponent of f and
6 + 2σ
K
σ
be a fixed positive constant. Now we define a configuration space:
Λ := (Q1 , . . . , QK ) ∈ Ω K ϕK (Q1 , . . . , QK ) M| ln |
9
10
(2.26)
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
13
14
(2.27)
43
44
45
15
16
where ϕK is defined at (2.14).
Let w be the unique solution of (2.17). By the well-known result of Gidas, Ni and Nirenberg [22], w is radially symmetric: w(y) = w(|y|) and strictly decreasing: w (r) < 0 for
r > 0, r = |y|. Moreover, we have the following asymptotic behavior of w:
N−1
1
,
w(r) = AN r − 2 e−r 1 + O
r
1
− N−1
−r
1+O
,
(2.28)
w (r) = −AN r 2 e
r
17
for r large, where AN > 0 is a constant. Let K(r) be the fundamental solution of − + 1
centered at 0. Then we have
1
w(r) = A0 + O
K(r),
r
1
K(r), for r 1,
(2.29)
w (r) = −A0 + O
r
27
K
Qj
u=
+φ
w z−
18
19
20
21
22
23
24
25
26
28
29
30
31
32
33
34
35
where A0 is a positive constant.
The idea of LEM is to look for solutions of (2.16) of the following type:
36
37
38
(2.30)
j =1
41
42
11
12
16
17
6
7
Ω
M>
3
4
10
11
1
Ω
5
6
497
39
40
41
where φ is solved first by Lyapunov–Schmidt reduction process, and (Q1 , . . . , QK ) are
adjusted so as to achieve a solution. LEM is a method of reducing the infinite-dimensional
problem of finding a critical point of J˜ to a finite-dimensional problem of (Q1 , . . . , QK ).
In general, it consists of the following five steps:
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J. Wei
S TEP 1. Find out good approximate functions.
1
2
3
4
5
6
7
8
2
This step contains most of the important computations. The idea is to choose good approximate functions such that the error S is small.
For Q ∈ Ω, we define w,Q to be the unique solution of
∂v
= 0 on ∂Ω .
∂ν
(2.31)
w,Q =
K
12
(2.32)
w,Qj .
19
15
We first analyze w,Q . To this end, set
16
x
|x − Q|
− w,Q
.
ϕ,Q (x) = w
ε
17
18
19
20
21
22
20
We state the following useful lemmas on the properties of ϕ,Q , whose proof can be
found in [44].
23
24
25
26
27
L EMMA 2.6. Assume that
have
M
2 | ln |
|x
ϕ,Q = − A0 + o(1) K
28
29
30
31
32
33
34
37
− Q∗ |
Ω,j
Qj 1
ϕK (Q) ,
= z −
2
Ω,K+1 = Ω
40
K
+O 2M+N +1
45
23
24
27
28
29
30
31
32
33
34
35
j = 1, . . . , K,
36
37
38
Ω,j .
(2.34)
39
40
j =1
41
L EMMA 2.7. For z ∈ Ω,j , j = 1, . . . , K, we have
43
44
22
26
(2.33)
41
42
21
25
√
The next lemma analyze w,Q in Ω . To this end, we divide Ω into K + 1-parts:
38
39
d(Q, ∂Ω) δ where δ is sufficiently small. We
where K(r) is the (radially symmetric) fundamental solution of − + 1 in RN , Q∗ =
Q + 2d(Q, ∂Ω)νQ̄ , νQ̄ denotes the unit outer normal at Q̄ ∈ ∂Ω and Q̄ is the unique
point on ∂Ω such that d(Q̄, Q) = d(Q, ∂Ω).
35
36
13
14
15
18
8
10
j =1
14
17
7
11
12
16
5
9
Let Q = (Q1 , . . . , QK ) ∈ Λ. We then define the approximate solution as
11
13
4
6
Q
= 0 in Ω ,
v − v + f w · −
9
10
3
w,Q = w,Qj
M
M
Qj
2
+ O K 2 .
=w z−
+ O K
42
43
44
(2.35)
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Existence and stability of spikes
1
499
For z ∈ Ω,K+1 , we have
2
1
M
w,Q = O K 2 .
3
4
2
(2.36)
4
5
6
7
5
w,Qk
9
10
12
21
22
23
14
w,Qk
|Q1 −Q2 |
36
37
38
39
40
43
44
45
19
21
22
23
24
(2.37)
25
26
27
28
γ0 =
RN
f w(y) e−y1 dy.
R EMARK . Note that γ0 > 0. See Lemma 4.7 of [61].
29
(2.38)
30
31
32
33
P ROOF. By (2.28), we have for |y| |Q1 − Q2 |,
N−1
2
Q1 −Q2
Q1 − Q2
= AN + o(1)
e−|y+ |
w y+
|y + Q1 − Q2 |
Q1 −Q2
|Q1 − Q2 | −y, |Q
+o(|y|)
1 −Q2 |
e
=w
.
41
42
18
20
where
33
35
large, it holds
Q2
Q1
|Q1 − Q2 |
w z−
= γ0 + o(1) w
f w z−
RN
31
34
16
29
32
15
17
L EMMA 2.8. For
26
30
12
Next we state a useful lemma about the interactions of two w’s.
25
28
11
M
= O K 2
which proves (2.35). The proof of (2.36) is similar.
24
27
10
k=j
17
20
9
13
16
19
8
and so
15
18
7
Qk
− ϕ,Qk (z)
=w z−
Q∗
Qk
M
k
= O e−|z− | + e−|z− | + M+N +1 = O 2
11
14
6
P ROOF. For k = j and z ∈ Ω,j , we have
8
13
3
34
35
36
37
38
39
40
41
Thus by Lebesgue’s Dominated Convergence Theorem
Q2
Q1
w z−
f w z−
RN
42
43
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J. Wei
Q1 − Q2
f w(y) w y +
RN
−y, Q1 −Q2 |Q1 − Q2 |
|Q1 −Q2 |
= 1 + o(1) w
f w(y) e
dy
RN
|Q1 − Q2 |
.
= γ0 + o(1) w
1
=
2
3
4
5
6
8
9
8
Let us define several quantities for later use:
10
11
12
9
10
B (Qj ) = −
Ω
f (wj )ϕ,Qj , B (Qi , Qj ) =
f (wi )wj .
(2.39)
Ω
13
14
17
18
19
20
21
22
25
26
27
15
L EMMA 2.9. For Q = (Q1 , . . . , QK ) ∈ Λ, it holds
16
2d(Qj , ∂Ω)
+ o w M| ln | ,
B (Qj ) = γ0 + o(1) w
|Qi − Qj |
+ o w M| ln | .
B (Qi , Qj ) = γ0 + o(1) w
32
33
34
35
36
37
38
39
40
43
44
45
18
19
20
21
(2.41)
22
24
25
26
27
29
Q∗j − Qj
√
+ O 2M+N +1
f (wj )w z −
Ω
|Qj − Q∗j |
+ o w(M| ln |)
+ o(1) w
2d(Qj , ∂Ω)
+ o w M| ln | .
+ o(1) w
B (Qj ) = 1 + o(1)
= γ
= γ
(2.40) follows from Lemma 2.6. To prove (2.41), we note that
41
42
(2.40)
28
and by Lemma 2.6
30
31
17
23
P ROOF. Note that
|x − Q∗ |
|x − Q∗ |
= 1 + o(1) w
A0 K
28
29
12
14
23
24
11
13
Then we have
15
16
7
B (Qi , Qj ) =
RN
−
Qi − Qj
Qi − Qj
f (w)w y −
f (w)w y −
RN \Ω,Qi
30
31
32
33
34
35
36
37
38
39
40
41
42
43
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Existence and stability of spikes
501
d(Qj ,∂Ω) σ d(Qi ,∂Ω)
|Qi − Qj |
+ O e−(1+ 2 ) e− = γ + o(1) w
|Qi − Qj |
+ o w M| ln | .
= γ + o(1) w
1
2
3
4
5
1
2
3
6
7
8
11
J˜
14
K
w,Qj
i=1
15
11
12
1
B (Qi )
= KI [w] −
2
K
17
18
13
14
i=1
1
−
2
16
15
B (Qi , Qj ) + o w M| ln | ,
(2.42)
i,j =1,...,K, i=j
23
24
20
K
w,Qj S j =1
21
CK
q+1
q +σ
22
M(1+σ )
2
(2.43)
for any q >
26
N
2.
27
28
28
The proof of Lemma 2.10 is technical and tedious. We refer to [44] for the computations.
30
31
S TEP 2. Obtain a priori estimates for a linear problem.
32
33
34
35
36
33
This is the fundamental step in reducing an infinite-dimensional problem to finitedimensional one. The key result we need here is the non-degeneracy assumption (f2).
Fix Q ∈ Λ. We define the following functions
37
40
41
Zi,j
∂wi
= ( − 1)
χi (z) ,
∂zj
2|z − Qi |
,
where χi (z) = χ
(M − 1)| ln |
44
45
35
36
i = 1, . . . , K, j = 1, . . . , N,
38
39
40
(2.44)
42
43
34
37
38
39
29
30
31
32
23
24
Lq (Ω )
25
26
29
17
19
and
25
27
16
18
19
22
8
10
L EMMA 2.10. For any Q = (Q1 , . . . , QK ) ∈ Λ and sufficiently small we have
13
21
7
9
12
20
5
6
We then have the following which provides the key estimates on the energy expansion
and error estimates.
9
10
4
41
42
where χ(t) is a smooth cut-off function such that χ(t) = 1 for |t| < 1 and χ(t) = 0 for
2
|t| > MM2 −1 . Note that the support of Zi,j belongs to B M 2 −1
( Q i ).
2M
| ln |
43
44
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J. Wei
In this step, we consider the following linear problem: Given h ∈ L2 (Ω ), find a function
φ satisfying
⎧
⎪
⎨ L [φ] := φ − φ + f (w,Q )φ = h + k,l ck,l Zk,l ;
φ, Zi,j = 0, i = 1, . . . , K, j = 1, . . . , N, and
(2.45)
⎪
⎩ ∂φ = 0 on ∂Ω ,
∂ν
12
13
14
15
16
17
18
19
20
φ∗ = φW 2,q (Ω ) ,
f ∗∗ = f Lq (Ω ) ,
(2.46)
27
28
29
30
31
32
33
34
35
36
37
38
39
42
43
44
45
6
7
11
where q > is a fixed number.
We have the following result:
13
P ROPOSITION 2.11. Let φ satisfy (2.45). Then for sufficiently small and Q ∈ Λ, we have
16
14
15
φ∗ Ch∗∗
(2.47)
17
18
19
where C is a positive constant independent of , K and Q ∈ Λ.
20
21
22
P ROOF. Arguing by contradiction, assume that
23
φ∗ = 1;
h∗∗ = o(1).
(2.48)
24
25
We multiply (2.45) by
∂wi
∂zj χi (z)
and integrate over Ω to obtain
∂wi
ck,l Zk,l ,
χi (z)
∂zj
k,l
∂wi
∂wi
= − h,
χi (z) + φ − φ + f (w,Q )φ,
χi (z) .
∂zj
∂zj
26
27
28
29
30
31
(2.49)
Observe that
∂wi
∂zj χi (z)
34
35
36
37
38
39
satisfies
40
∂wi
∂wi
∂wi
χi (z) −
χi (z) + f (wi )
χi (z)
∂zj
∂zj
∂zj
= 2∇z
∂wi
∂wi
∇z χi + (χi )
.
∂zj
∂zj
32
33
From the exponential decay of w one finds
∂wi
χi (z) = o(1).
h,
∂zj
40
41
5
12
N
2
25
26
4
10
23
24
3
9
21
22
2
8
for some constants ck,l , k = 1, . . . , K, l = 1, . . . , N .
To this purpose, we define two norms
10
11
1
41
42
43
44
(2.50)
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Existence and stability of spikes
1
2
3
4
5
6
7
8
9
Integrating by parts and using Lemma 2.7, we deduce
12
13
14
1
2
∂wi
χi (z)
φ − φ + f (w,Q )φ,
∂zj
M−1
∂wi
= f (w,Q ) − f (wi )
χi (z), φ + O 2 φ∗
∂zj
σ Mσ
= O K 2 φ∗ = o φ∗ = o(1)
10
11
503
where we have used the fact that M >
6+2σ
σ N
f (w,Q ) − f (wi ) ∂wi χi ∂z
j
∗∗
3
4
5
6
7
8
9
10
and that
11
Mσ
σ ∂wi C |w,Q − wi | χi K σ 2 .
∂z
j
∗
15
16
17
19
17
(2.51)
20
24
26
27
21
∂wi
Zk,l ,
χi (z) = 0
∂zj
22
(2.52)
29
25
26
∂wi
Zi,l ,
χi (z) = O M .
∂zj
27
(2.53)
33
30
The left hand side of (2.49) becomes
ci,j +
31
O M ci,l = o(1)
32
33
l=j
34
34
35
36
37
38
35
and hence
ci,j = o(1),
36
37
i = 1, . . . , K, j = 1, . . . , N.
(2.54)
39
40
φi = φχi ,
where
χi
43
44
45
38
39
To obtain a contradiction, we define the following cut-off functions:
41
42
28
29
30
32
23
24
and for k = i and l = j , we have
28
31
18
19
On the other hand, for k = i we have
23
25
14
16
∂wi
∂w 2
Zi,j ,
χi (z) = −
f (w)
dy + o(1).
∂zj
∂yj
RN
20
22
13
15
It is easy to see that
18
21
12
Note that χi = 1 for z ∈ B M 2 −1
2M
40
2|z − Qi |
, i = 1, . . . , K.
=χ
(M − M −1 )| ln |
| ln |
41
(2.55)
42
43
( Q i ) and the support of φ belongs to B M | ln | ( Q i ).
2
44
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J. Wei
Then the conditions φ, Zi,j = 0 is equivalent to
1
2
3
2
φi , Zi,j = 0.
(2.56)
4
5
4
5
The equation for φi becomes
6
7
8
φi − φi + f (w,Q )φi =
11
12
13
6
ci,j Zi,j
+ hχi + 2∇φ∇χi + χi φ.
(2.57)
j
9
10
16
11
f (w,Q )φi = f (wi ) + o M/2−N φi .
12
(2.58)
q
φi W 2,q (Ω
)
q
C hχ q
i L (Ω )
q
+ C 2∇φ∇χ + χ φ q
i
i
L (Ω
(2.59)
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
20
φ=
22
φi + Φ
(2.60)
where Φ = φ(1 −
44
45
23
24
i=1
25
K
i=1 χi ). Then the equation for Φ becomes
26
27
Φ − Φ + f (w,Q )Φ
K
K
K
=h 1−
χi − 2
∇φ∇χi −
χi φ.
i=1
i=1
28
29
30
(2.61)
i=1
33
By Lemma 2.7, f (w,Q )Φ = o(1)Φ. Standard regularity theorem gives
K
q
ΦW 2,q (Ω ) C h 1 −
χi
i=1
34
q
q
35
36
37
L (Ω )
K
q
K
+ C 2
∇φ∇χi +
(χi )φ q
i=1
31
32
i=1
L Ω
38
39
.
(2.62)
42
43
18
21
K
25
26
16
19
Next, we decompose
22
24
15
17
.
)
21
23
13
14
Using (2.56) and (2.58), a contradiction argument similar to that of Proposition 3.2 of
[27] gives
19
20
8
10
Lemma 2.7 yields
17
18
7
9
14
15
3
40
41
42
Lp -regularity
is independent of < 1. The case of
(Observe that the constant C in the
Dirichlet boundary condition has been proved in Lemma 6.4 of [61]. The case of Neumann
boundary condition can be proved similarly.)
43
44
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Existence and stability of spikes
1
Combining (2.60), (2.59) and (2.62), we obtain
2
4
i=1
5
6
C
7
8
K
5
6
q
φi W 2,q (Ω )
q
+ CΦW 2,q (Ω )
i=1
12
7
8
i=1
q
q
9
10
11
L (Ω )
12
K
2∇φ∇χ + (χ )φ q q
+C
13
14
i
i=1
15
16
17
22
4
)
K
K
q
hχ q
h
1
−
C
+
χi
i L (Ω )
11
21
W 2,q (Ω )
i=1
10
20
3
q
+ CΦW 2,q (Ω
9
19
1
2
K q
q
φW 2,q (Ω ) C φi 3
18
505
q
ChLq (Ω )
i
13
14
L (Ω )
q
+ O | ln |−1 φW 2,q (Ω
15
16
)
17
18
since
K
K
q
χi + 1 −
χi
i=1
19
q
2,
−1
|∇χ | + |χ | C | ln | .
20
(2.63)
i=1
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
23
This gives
24
φW 2,q (Ω ) = o(1).
L [φ] = h +
(2.65)
ci,j Zi,j ,
i,j
φ, Zi,j = 0,
i = 1, . . . , K, j = 1, . . . , N,
∂φ
= 0 on ∂Ω .
∂ν
28
31
32
33
34
35
36
38
(2.66)
39
40
41
Moreover, we have
42
42
φ∗ Ch∗∗
44
45
27
37
40
43
26
30
P ROPOSITION 2.12. There exists 0 > 0 such that for any 0 < < 0 the following property holds true. Given h ∈ W 2,q (Ω ), there exist s a unique pair (φ, c) =
(φ, {ci,j }i=1,...,K,j =1,...,N ) such that
25
29
From Proposition 2.11, we derive the following existence result:
38
41
(2.64)
A contradiction to (2.48).
37
39
21
(2.67)
43
44
for some positive constant C.
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J. Wei
P ROOF. The bound in (2.67) follows from Proposition 2.11 and (2.54). Let us now prove
the existence part. Set
1
where we define the inner product on
12
13
14
15
16
17
(u, v) =
5
6
as
7
8
(∇u∇v + uv).
9
Ω
10
11
Note that, integrating by parts, one has
ψ ∈H
if and only if
12
ψ, Zi,j = 0,
13
i = 1, . . . , K, j = 1, . . . , N.
14
15
Observe that φ solves (2.65) and (2.66) if and only if φ ∈ H satisfies
18
19
)
Ω
16
17
!
"
(∇φ∇ψ + φψ) − f (w,Q )φ, ψ = h, ψ ,
∀ψ ∈ H.
18
19
20
21
20
This equation can be rewritten as
21
22
23
22
φ + S(φ) = h̄ in H,
(2.68)
24
25
26
27
28
35
36
37
38
39
40
41
42
43
44
45
26
27
28
30
31
φ = A (h).
(2.69)
33
34
25
29
In the following, if φ is the unique solution given in Proposition 2.12, we set
31
32
23
24
where h̄ is defined by duality and S : H → H is a linear compact operator.
Using Fredholm’s alternative, showing that equation (2.68) has a unique solution for
each h̄, is equivalent to showing that the equation has a unique solution for h̄ = 0, which
in turn follows from Proposition 2.11 and our proof is complete.
29
30
2
4
10
11
H 1 (Ω
1
3
H = u ∈ H (Ω ) u, ( − 1)−1 Zi,j = 0
8
9
F:chipot506.tex; VTEX/Rita p. 20
32
33
34
Note that (2.67) implies
35
A (h) Ch∗∗ .
∗
(2.70)
37
38
S TEP 3. A non-linear Lyapunov–Schmidt reduction.
39
For small and for Q ∈ Λ, we are going to find a function φ,Q such that for some
constants ci,j , j = 1, . . . , N , the following equation holds true
(w,Q + φ) − (w,Q + φ) + f (w,Q + φ) =
φ, Zi,j = 0,
j = 1, . . . , N,
36
∂φ
∂ν
k,l ck,l Zk,l
= 0 on ∂Ω .
40
41
42
43
in Ω ,
(2.71)
44
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Existence and stability of spikes
1
507
The first equation in (2.71) can be written as
2
1
2
φ − φ + f (w,Q )φ = −S [w,Q ] + N [φ] +
ci,j Zi,j ,
3
4
3
4
i,j
5
6
7
8
5
where
6
N [φ] = − f (w,Q + φ) − f (w,Q ) − f (w,Q )φ .
7
(2.72)
9
10
11
12
13
14
9
L EMMA 2.13. For Q ∈ Λ and sufficiently small, we have for φ∗ + φ1 ∗ + φ2 ∗ 1,
N [φ]
17
18
19
20
21
22
23
N [φ1 ] − N [φ2 ]
(2.73)
C φ1 σ∗ + φ2 σ∗ φ1 − φ2 ∗ .
∗∗
26
27
28
29
20
21
f (w,Q + φ) − f (w,Q ) − f (w,Q )φ| C φ|1+σ ,
22
23
P ROPOSITION 2.14. For Q ∈ Λ and sufficiently small, there exists a unique φ = φ,Q
such that (2.71) holds. Moreover, Q → φ,Q is of class C 1 as a map into W 2,q (Ω ) ∩ H,
and we have
φ,Q ∗ rK
q+1
q +σ
34
35
36
37
40
41
(2.75)
44
45
25
26
27
28
29
31
32
for some constant r > 0.
33
34
P ROOF. Let A be as defined in (2.69). Then (2.71) can be written as
φ = A −S [w,Q ] + N [φ] .
35
36
(2.76)
37
38
Let r be a positive (large) number, and set
39
q+1
)
+σ M(1+σ
2
.
Fr = φ ∈ H ∩ W 2,q (Ω ): φ∗ < rK q
42
43
24
30
M(1+σ )
2
38
39
17
19
Since f is Hölder continuous with exponent σ , we deduce
which implies (2.73). The proof of (2.74) goes along the same way.
16
18
f (w,Q + φ) − f (w,Q ) = f (w,Q + θ φ)φ.
32
33
14
15
30
31
12
13
(2.74)
P ROOF. Inequality (2.73) follows from the mean-value theorem. In fact, for all z ∈ Ω
there holds
24
25
10
11
Cφ∗1+σ ;
∗∗
15
16
8
40
41
42
Define now the map G : Fr
→ H ∩ W 2,q (Ω
)
G [φ] = A −S [w,Q ] + N [φ] .
as
43
44
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7
8
9
10
11
12
13
14
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J. Wei
Solving (2.71) is equivalent to finding a fixed point for G . By Lemmas 2.10 and 2.13, for
sufficiently small and r large we have
4
5
G [φ1 ] − B [φ2 ] C N [φ1 ] − N [φ2 ] < 1 φ1 − φ2 ∗ ,
∗
∗
2
6
7
which shows that G is a contraction mapping on Fr . Hence there exists a unique φ =
φ,Q ∈ Fr such that (2.71) holds.
Now we come to the differentiability of φ,Q . Consider the following map H : Λ × H ∩
W 2,q (Ω ) × R N K → H ∩ W 2,q (Ω ) × R N K of class C 1
⎜
⎜
⎜
H (Q, φ, c) = ⎜
⎜
⎜
⎝
( − 1)−1 (S
[w,Q
+ φ]) −
−1
i,j ci,j ( − 1) Zi,j
(φ, ( − 1)−1 Z1,1 )
..
.
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
43
44
45
9
10
11
12
14
⎟
⎟
⎟
⎟.
⎟
⎟
⎠
15
16
17
18
(φ, ( − 1)−1 ZK,N )
19
(2.77)
20
21
Equation (2.71) is equivalent to H (Q, φ, c) = 0. We know that, given Q ∈ Λ, there is
a unique local solution φ,Q , c,Q obtained with the above procedure. We prove that the
linear operator
∂H (Q, φ, c) : H ∩ W 2,q (Ω ) × R N K → H ∩ W 2,q (Ω ) × R N K
∂(φ, c) (Q,φ,Q ,c,Q )
22
is invertible for small. Then the
of Q → (φ,Q , c,Q ) follows from the
Implicit Function Theorem. Indeed we have
∂H (Q, φ, c) [ψ, d]
∂(φ, c) (Q,φ,Q ,c,Q )
⎛
⎞
( − 1)−1 (S [w,Q + φ,Q ](ψ)) − i,j dij ( − 1)−1 Zi,j
⎜
⎟
⎜
⎟
(ψ, ( − 1)−1 Z1,1 )
⎜
⎟
⎟.
=⎜
⎜
⎟
..
⎜
⎟
.
⎝
⎠
−1
(ψ, ( − 1) ZK,N )
29
C 1 -regularity
41
42
8
13
⎞
21
22
2
3
q+1
M(1+σ )
G [φ] C S [w,Q ] + C N [φ] < rK q +σ 2 ,
∗
∗∗
∗∗
⎛
1
23
24
25
26
27
28
30
31
32
33
34
35
36
37
38
39
40
41
Since φ,Q ∗ is small, the same proof as in that of Proposition 2.11 shows that
∂H (Q, φ, c) ∂(φ, c) (Q,φ,Q ,c,Q )
42
43
44
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Existence and stability of spikes
1
2
509
is invertible for small.
This concludes the proof of Proposition 2.14.
1
3
4
5
6
3
In some cases (e.g., critical or nearly critical exponent problems), we need to obtain
further differentiability of φ,Q (e.g., C 2 in Q). This will be achieved by further reduction.
See [13,65] and [66] for such arguments.
7
8
11
14
15
16
17
18
19
20
21
22
Fix Q ∈ Λ. Let φ,Q be the solution given by Proposition 2.14. We define a new functional
27
28
29
30
31
32
M (Q) = J˜ [w,Q + φ,Q ] : Λ → R.
(2.78)
35
P ROOF. By Proposition 2.14, there exists 0 such that for 0 < < 0 we have a C 1 map
which, to any Q ∈ Λ, associates φ,Q such that
S [w,Q + φ,Q ] =
(2.79)
20
21
22
26
27
29
30
31
i = 1, . . . , K, j = 1, . . . , N.
32
33
Hence we have
Ω
34
∂(w,Q + φ,Q ) ∇u ∇
∂Q
i,j
∂(w,Q + φ,Q ) − f (u )
∂Q
i,j
∂(w,Q + φ,Q ) +u
∂Q
Qi =Qi
Qi =Qi
i,j
35
36
Qi =Qi
37
38
= 0,
39
40
41
45
19
28
for some constants ckl
Let Q ∈ Λ be a critical point of M . Set u = w,Q + φ,Q . Then we have
40
44
18
25
∈ R KN .
39
43
17
24
φ,Q , Zi,j = 0
38
42
14
23
ckl Zk,l ,
k=1,...,K; l=1,...,N
DQi,j |Qi =Qi M (Q ) = 0,
13
16
L EMMA 2.15. If Q is critical point of M (Q) in Λ, then u = w,Q + φ,Q is a critical
point of J˜ [u].
36
37
11
15
Then we have the following reduction lemma
33
34
10
12
25
26
6
9
23
24
5
8
S TEP 4. A reduction lemma.
12
13
4
7
9
10
2
41
which gives
42
k=1,...,K; l=1,...,N
ckl
∂(w,Q + φ,Q ) Zk,l
= 0.
∂Qi,j
Ω
Qi =Q
i
43
(2.80)
44
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J. Wei
We claim that (2.80) is a diagonally dominant system. In fact, since φ,Q , Zi,j = 0,
we have that
∂φ,Q
∂Zk,l
Zk,l
=
−
φ,Q
= 0 if k = i.
∂Qi,j
∂Qi,j
Ω
Ω
6
If k = i, we have
8
10
11
∂φ,Q
Zk,l
=−
∂Qk,j
Ω
Ω
∂Zk,l ∂Zk,l
φ,Q ∗∗
φ,Q = ∂Qk,j
∂Qk,j ∗∗
M(1+σ ) −( q+1 +σ )N −1 q+1 +σ M(1+σ ) −1 q
=O 2
=O K q
2
M
=O 2 .
13
14
15
17
18
23
24
25
Zk,l
∂w,Qi
∂Qi,j
Ω
=
Q
Ω ∩B M | ln | ( k
2
Zk,l
Ω
∂w,Qk
Zk,l
)
∂Qk,j
34
35
36
37
38
39
∂w,Qi
∂Qi,j
= O M .
42
=
Ω ∩B M | ln | (
2
= − −1 δlj
Qk
Zk,l
)
45
13
14
18
19
20
24
∂Qk,j
25
∂w 2
f (w)
+ O(1).
∂yj
RN
For each (k, l), the off-diagonal term gives
M M
M
O 2 +
+
O() = O 2 + K M + = o(1)
k=i
k=i,l=j
by our choice of M > 6+2σ
σ N.
Thus equation (2.80) becomes a system of homogeneous equations for ckl and the matrix
of the system is
non-singular. So ckl ≡ 0, k = 1, . . . , K, l = 1, . . . , N .
Hence u = K
i=1 w,Qi + φ,Q1 ,...,QK is a solution of (2.20).
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
S TEP 5. Using variational arguments to find critical points for the finite-dimensional reduced problem.
43
44
12
∂w,Qk
40
41
11
23
29
33
10
22
28
32
9
21
27
31
8
17
For k = i, we have
26
30
5
16
21
22
4
15
For k = i, we have
19
20
3
7
12
16
2
6
7
9
1
41
42
43
By Lemma 2.15, we just need to find a critical point for the reduced energy functional M (Q). Depending on the asymptotic behavior of the reduced energy functional,
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7
one can use either local minimization, or local maximization [29], or saddle point techniques [66]. Here there is no compactness problem since the reduced problem is already
finite-dimensional.
We first obtain an asymptotic formula for M (Q). In fact for any Q ∈ Λ, we have
˜
M (Q) = J [w,Q ] +
(∇w,Q ∇φ,Q + w,Q φ,Q )
8
−
9
Ω
10
12
16
K
22
23
24
15
16
17
18
19
20
21
22
23
(2.81)
26
27
28
P ROPOSITION 2.16. For small, the following maximization problem
34
35
36
37
38
39
29
max{M (Q): Q ∈ Λ}
31
33
has a solution
Q
∈ Λ◦ —the
(2.82)
42
31
33
P ROOF. First, we obtain a lower bound for M : Recall that KΩ (r) is the maximum number of non-overlapping balls with equal radius r packed in Ω. Now we choose K such
that
M + 2N
1 K KΩ
| ln | .
(2.83)
2
34
35
36
37
38
39
40
= (Q01 , . . . , Q0K ) be the centers of arbitrary
balls. Certainly Q0 ∈ Λ. Then we have
Q0
Let
| ln |)
43
44
45
30
32
interior of Λ.
40
41
24
25
We shall prove
30
32
7
14
28
29
6
13
26
27
5
12
i=j
25
4
11
1
2d(Qi , ∂Ω)
γ0 + o(1)
w
2
i=1
|Qi − Qj |
1
+ o w M| ln | .
w
− γ0 + o(1)
2
M (Q) = KI [w] −
21
3
10
−S [w,Q ] φ,Q + O φ,Q 2∗
by Lemma 2.10, Proposition 2.14 and the choice of M at (2.26).
By Lemma 2.10, we obtain
20
2
9
2+ 2 +2σ M(1+σ ) = J˜ [w,Q ] + O K q
= J˜ [w,Q ] + o w M| ln |
15
1
8
= J˜ [w,Q ] + O S [w,Q ]∗∗ φ,Q ∗ + O φ,Q 2∗
14
19
f (w,Q )φ,Q + O φ,Q 2∗
Ω
13
18
Ω
= J˜ [w,Q ] +
11
17
511
w
2d(Q0i , ∂Ω)
e
2d(Q0i ,∂Ω)
−
K balls among those
×
41
42
M+2N ,
KΩ ( M+2N
2
w
|Q0i − Q0j |
43
M+2N
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J. Wei
and hence
2
1
K
γ0 + o(1) M+2N
M Q M Q0 KI [w] −
2
2
K γ0 + o(1) M+2N + o w(M| ln |)
−
2
KI [w] − K 2 γ0 + o(1) M+2N + o w M| ln | .
3
4
5
6
7
8
2
3
4
5
6
7
(2.84)
9
10
11
12
9
On the other hand, if Q ∈ ∂Λ, then either there exists (i, j ) such that |Qi − Qj | =
M| ln |, or there exists a k such that d(Qk , ∂Ω) = M
2 | ln |. In both cases we have
1
M Q KI [w] − γ0 + o(1) w M| ln | + o w M| ln | .
2
13
14
15
16
(2.86)
33
34
37
38
39
40
Let p =
30
By suitable scaling, (2.4) becomes the following problem
31
32
N+2
N−2
= 0 in Ω,
u − μu + u
u > 0 in Ω and ∂u
∂ν = 0 on ∂Ω
(2.87)
45
33
34
35
where μ = is large.
It is well known that the solutions to
36
1
2
37
38
N+2
U + U N−2 = 0
(2.88)
39
40
41
are given by the following
43
44
25
29
N +2
N −2 .
41
42
24
28
35
36
22
27
2.4. Bubbles to (2.4): the critical case
29
32
21
26
27
31
19
23
C OMPLETION OF P ROOF OF T HEOREM 2.5. Theorem 2.5 follows from Proposition 2.16
and the reduction Lemma 2.15.
26
30
18
20
which is impossible.
We conclude that Q ∈ Λ. This completes the proof of Proposition 2.16.
23
28
14
17
20
25
12
16
−2N
w M| ln | 2K 2 M+2N C M | ln |
19
24
11
15
Combining (2.85) and (2.84), we obtain
18
22
10
13
(2.85)
17
21
8
UΛ,ξ = cN
1
Λ2 + |x − ξ |2
42
43
N−2
2
,
where Λ > 0, ξ ∈ RN .
44
(2.89)
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8
9
10
11
12
13
14
15
16
513
4
N−2
+2
A notable difference here is that the linearized operator + ( N
N −2 )UΛ,ξ has (N + 1)dimensional kernels. Namely,
4
∂Uλ,ξ ∂UΛ,ξ
∂Uλ,ξ
N + 2 N−2
.
UΛ,ξ = span
,
,...,
Kernel +
N −2
∂Λ
∂ξ1
∂ξN
2
3
4
(2.90)
5
6
Thus when we apply LEM, we need also to take care of the scaling parameters. See
[13,43,65,66] and the references therein.
Concerning boundary bubbles, the existence of mountain-pass solutions was first proved
in Wang [69] and Adimurthi and Mancini [1]. Ni, Takagi and Pan [55] showed the least
energy solutions develop a bubble at the maximum point of the mean curvature (thereby
establishing results similar to Theorem 2.1). Local mountain-pass solutions concentrating
on one or separated boundary points are established in [23]. At non-degenerate critical
points of the positive mean curvature, single boundary bubbles exist [2]. Lin, Wang and Wei
[43] established results similar to Theorem 2.2 for dimension N 7, at a non-degenerate
local minimum point of the mean curvature with positive value:
17
18
1
7
8
9
10
11
12
13
14
15
16
17
18
T HEOREM 2.17. Suppose the following two assumptions hold:
19
19
20
(H1) N 7,
20
21
(H2) Q0 = 0 is a non-degenerate minimum point of H (Q) and H (Q0 ) > 0.
21
22
23
24
25
26
Let K 2 be a fixed integer. Then there exists a μK > 0 such that for μ > μK , problem
(2.87) has a non-trivial solution uμ with the following properties
(1)
27
28
29
30
31
32
33
34
35
36
37
u(x) =
K
j =1
U1
μ Λj ,Q0 +μ
R[Q̂1 , . . . , Q̂K ] := c1
K
j =1
where ϕ(Q) =
45
26
29
ϕ(Q̂j ) + c2
1
i=j
|Q̂i − Q̂j |N −2
k,l ∂k ∂l H (Q0 )Qk Ql , c1
31
32
33
34
35
36
37
38
(2.91)
39
40
41
and c2 are two generic constants.
43
44
25
30
40
42
24
28
Q̂j
(∗) Find out the optimal configuration (Q̂1 , . . . , Q̂K ) that minimizes the functional
R[Q̂1 , . . . , Q̂K ].
Here for Q̂ = (Q̂1 , . . . , Q̂K ) ∈ R (N −1)K , Q̂i = Q̂j , we define
41
23
27
N−4 2
,
μ +O μ
where Λj → Λ0 := A0 H (Q0 ) > 0, j = 1, . . . , K, and
μ
μ
(2) Q̂μ := (Q̂1 , . . . , Q̂K ) approach an optimal configuration in the following problem:
38
39
3−N
N
22
42
43
Theorem 2.17 is proved by LEM. Here the computation is more complicated, since the
interaction between bubbles is very involved.
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Concerning interior bubbles, under some assumptions, it is proved in [24] and [64] that
there are no interior bubble solutions. However interior bubble solutions can be recovered if
one add the boundary layers. (The boundary layer solution has been constructed in [50] (see
Section 2.6).) The following result establishes the existence of multiple interior bubbles in
dimension N = 3, 4, 5.
6
7
8
9
14
15
16
17
20
21
22
23
24
25
26
27
28
29
30
31
32
33
36
37
38
39
40
41
In the slightly supercritical case, we let p =
N +2
N −2
u − μu + up = 0 in Ω,
u > 0 in Ω and ∂u
∂ν = 0 on ∂Ω.
8
9
14
15
(2.92)
16
17
18
The following result was proved by [66] and [14] through the use of LEM.
19
T HEOREM 2.19. Let N 3. Then δ > 0 sufficiently small, problem (2.92) admits a boundary bubble solution.
In fact, in the slightly supercritical case, there is also the phenomena of bubble-towers.
A bubble-tower is a sum of bubbles centered at the same point
K
20
21
22
23
24
25
26
UΛj ,ξ ,
j =1
Λj +1
where Λ1 ,
→ +∞, j = 1, . . . , K − 1.
Λj
(2.93)
27
28
29
This has been discussed in [15] and [25].
It is completely open whether or not point condensation solutions exist for (2.92) when
+2
p> N
N −2 + δ. In fact, let Ω be the unit ball. Using Pohozaev’s identity, it is not difficult to
show that there exists a positive constant c0 , independent of 1, such that
30
31
32
33
34
inf u c0
Ω
(2.94)
for all radial solution u of (2.4). This marks a basic difference between the behavior of
+2
N +2
solutions of these two cases p N
N −2 and p > N −2 . It eliminates the possibility of the
existence of a radial spiky solution which approaches zero in measure as approaches
+2
zero in the supercritical case p > N
N −2 .
35
36
37
38
39
40
41
42
2.6. Concentration on higher-dimensional sets
44
45
7
13
+ δ where δ > 0. Consider
42
43
5
12
34
35
4
11
2.5. Bubbles to (2.4): slightly supercritical case
18
19
3
10
11
13
2
6
T HEOREM 2.18. (See [71,92].) Let N = 3, 4, 5. For any fixed integer k, then problem
(2.87) has a solution (at least along a subsequence k → 0) with k interior bubbles and
one boundary layer.
10
12
1
43
44
The following conjecture has been made by Ni [53,54].
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1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
Fig. 1. Lines intersecting with ∂Ω orthogonally.
12
12
13
14
15
16
13
C ONJECTURE . Given any integer 0 k n − 1, there exists pk ∈ (1, ∞) such that for
1 < p < pk , (2.4) possesses a solution with k-dimensional concentration set, provided that
is sufficiently small.
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
15
16
17
Progress in this direction has only been made very recently. In [49] and [50], Malchiodi
and Montenegro proved that for N 2, there exists a sequence of numbers εk → 0 such
that problem (2.4) has a solution uεk which concentrates at boundary of ∂Ω (or any component of ∂Ω). Such a solution has the following energy bound
(2.95)
In [48], Malchiodi showed the concentration phenomena for (2.4) along a closed nondegenerate geodesic of ∂Ω in three-dimensional smooth bounded domain Ω. F. Mahmoudi and A. Malchiodi in [51] prove a full general concentration of solutions along
k-dimensional (1 k n − 1) non-degenerate minimal sub-manifolds of the boundary
for n 3 and 1 < p < n−k+2
n−k−2 . When Ω = B1 (0), there are also multiple (radially symmetric) clustered interfaces near the boundary [52].
For concentrations on lines intersecting with the boundary, Wei and Yang [93] made
the first attempt in the two-dimensional case. Let Γ ⊂ Ω ⊂ R2 be a curve satisfying the
following assumptions: The curvature of Γ is zero and Γ intersects ∂Ω at exactly two
points, saying, γ1 , γ0 and at these points Γ ⊥∂Ω. Let −k1 and k0 are the curvatures of the
boundary ∂Ω at the points γ1 and γ0 respectively. A picture of Γ and Ω is as follows:
We define a geometric eigenvalue problem
−f (θ ) = λf (θ ),
18
19
20
21
22
Jεk [uεk ] ∼ εkN −1 .
f (0) + k0 f (0) = 0.
24
25
26
27
28
29
30
31
32
33
34
35
36
38
f (1) + k1 f (1) = 0,
23
37
0 < θ < 1,
39
(2.96)
40
41
We say that Γ is non-degenerate if (2.96) does not have a zero eigenvalue. This is equivalent to the following condition:
44
45
14
42
43
44
k0 − k1 + k0 k1 |Γ | = 0,
(2.97)
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J. Wei
where |Γ | denotes the length of Γ .
Moreover, we set up the gap condition that there exists a small constant c > 0
2 2 λ0 − k π ε 2 cε,
|Γ |2 1
2
3
∀k ∈ N.
(2.98)
6
7
10
11
12
13
14
15
16
7
8
T HEOREM 2.20. We assume that the line segment Γ satisfies the non-degenerate condition (2.97). Given a small constant c, there exists ε0 such that for all ε < ε0 satisfying the
gap condition (2.98), problem (2.4) has a positive solution uε concentrating along a curve
Γ near Γ . Moreover, there exists some number c0 such that uε satisfies globally,
uε (x) exp −c0 ε −1 dist(x, Γ )
19
20
21
22
23
24
25
26
27
28
29
30
31
34
10
11
12
13
14
16
17
R EMARK 2.6.1. The geometric eigenvalue problem (2.96) was first introduced by M.
Kowalczyk in [37] where he constructed layered solution concentrating on a line for the
Allen–Cahn equation.
R EMARK 2.6.2. Theorem 2.20 is proved using the infinite-dimensional Lyapunov–
Schmidt reduction technique introduced in [18].
R EMARK 2.6.3. One can also constructed multiple clustered line concentrating solutions,
using the Toda system. See [94]. This follows from earlier work in [19], where multiple
clustered interfaces are constructed at non-minimizing lines for the Allen–Cahn equation.
It is quite interesting to see the connection between Toda system
qj
+e
qj −qj +1
−e
qj −1 −qj
=0
(2.99)
18
19
20
21
22
23
24
25
26
27
28
29
30
31
and clustered interfaces.
32
33
9
15
and the curve Γ will collapse to Γ as ε → 0.
17
18
5
6
In [93], the following result was proved
8
9
4
32
R EMARK 2.6.4. It will be interesting to construct solutions concentrating on surfaces
which intersect with ∂Ω orthogonally.
33
34
35
35
36
36
37
37
2.7. Robin boundary condition
38
39
40
41
42
43
44
45
38
Robin boundary conditions are particularly interesting in biological models where they
often arise. We refer the reader to [10] for this aspect.
In [3], Berestycki and Wei discussed the existence and asymptotic behavior of least energy solution for following singularly perturbed problem with Robin boundary condition:
39
40
41
42
43
2 u − u + up
∂u
∂ν
= 0, u > 0 in Ω,
+ λu = 0
on ∂Ω,
(2.100)
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517
where λ > 0. Similar to [57], we can define the following energy functional associated with
(2.100):
1
2
1
λ
J [u] :=
|∇u|2 +
u2 −
F (u) +
u2 ,
2 Ω
2 Ω
2
Ω
∂Ω
u
where F (u) = 0 f (s) ds, f (s) = s p , u ∈ H 1 (Ω).
Similarly, for ∈ (0, 1), we can define the so-called mountain-pass value
c,λ = inf max J h(t)
3
h∈Γ 0t1
(2.101)
13
14
15
16
17
18
19
8
9
(2.102)
where Γ = {h:
is continuous, h(0) = 0, h(1) = e}.
For fixed small, as λ moves from 0 (which is Neumann BC) to +∞ (which is Dirichlet
BC), by the results of [57,58] and [61], the asymptotic behavior of u,λ changes dramatically: a boundary spike is displaced to become an interior spike. The question we shall
answer is: where is the borderline of λ for spikes to move inwards?
Note that when N = 1, by ODE analysis, it is easy to see that the borderline is exactly
at λ = 1. In fact, we may assume that Ω = (0, 1), and a s → 0, the least energy solution
converges to a homoclinic solution of the following ODE:
w(y) → 0 as |y| → +∞.
(2.103)
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
13
14
15
16
17
18
19
21
22
Then it follows that
23
(w )2 = w 2 −
26
27
12
20
w − w + w p = 0 in R1 ,
24
25
10
11
[0, 1] → H 1 (Ω) | h(t)
22
23
5
7
20
21
4
6
11
12
2
2
w p+1 ,
p+1
24
|w | < w.
(2.104)
26
w (0) − λw(0)
= 0. We see
As → 0, the limiting boundary condition (2.100) becomes
from (2.104) that this is possible if and only if λ < 1.
When N = 2, the situation changes dramatically. To understand the location of the spikes
at the boundary, an essential role is played by the analogous problem in a half space with
Robin boundary condition on the boundary. Thus we first consider
u − u + f (u) = 0, u > 0 in RN
+,
(2.105)
∂u
N
1
u ∈ H (R+ ), ∂ν + λu = 0 on ∂RN
+
N
where RN
+ = {(y , yN ) | yN > 0} and ν is the outer normal on ∂R+ .
Let
1
1 2
λ
2
Iλ [u] =
F (u) +
u2 .
|∇u| + u −
+
+
2
2
2
R+
R
∂R
N
N
N
inf
v≡0, v∈H 1 (R+
N)
sup Iλ [tv].
t>0
27
28
29
30
31
32
33
34
35
36
37
38
39
(2.106)
40
41
42
As before, we define a mountain-pass vale for Iλ :
cλ =
25
43
(2.107)
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J. Wei
Our first result deals with the half space problem:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
2
T HEOREM 2.21.
(1) For λ 1, cλ is achieved by some function wλ , which is a solution of (2.105).
(2) For λ large enough, cλ is never achieved.
(3) Set
λ∗ = inf{λ | cλ is achieved}.
26
27
28
31
32
The proof of Theorem 2.21 is by the method of concentration-compactness, and the
method of vanishing viscosity.
Now consider the problem in a bounded domain.
T HEOREM 2.22. Let λ λ∗ and u,λ be a least energy solution of (2.100). Let x ∈ Ω
be a point where u,λ reaches its maximum value. Then after passing to a subsequence,
x → x0 ∈ ∂Ω and
(1) d(x , ∂Ω)/ → d0 , for some d0 > 0,
(2) v,λ (y) = u,λ (x + y) → wλ (y) in C 1 locally, where wλ attains cλ of (2.107) (and
thus is a solution of (2.105)),
(3) the associated critical value can be estimated as follows:
c,λ = N cλ − H̄ (x0 ) + o()
(2.109)
37
38
39
40
41
42
43
44
45
8
11
12
13
14
15
16
17
18
19
20
21
22
23
24
where cλ is given by (2.107), and H̄ (x0 ) is given by the following
∂wλ
y · ∇ wλ
H (x0 )
H̄ (x0 ) = max −
∂yN
wλ ∈Sλ
R+
N
25
26
(2.110)
27
28
29
where Sλ is the set of all solutions of (2.105) attaining cλ , and y = (y1 , . . . , yN −1 ),
∂
∇ = ( ∂y∂ 1 , . . . , ∂yN−1
),
(4) H̄ (x0 ) = maxx∈∂Ω H̄ (x).
30
31
32
33
On the other hand, when λ > λ∗ , a different asymptotic behavior appears.
35
36
6
10
33
34
5
9
Then λ∗ > 1 and for λ λ∗ , cλ is achieved, and for λ > λ∗ , cλ is not achieved.
29
30
4
7
(2.108)
24
25
3
34
35
T HEOREM 2.23. Let λ > λ∗ and u,λ be a least energy solution of (2.100). Let x ∈ Ω
be a point where u,λ reaches its maximum value. Then after passing a subsequence, we
have
(1) d(x , ∂Ω) → maxx∈Ω d(x, ∂Ω),
(2) v,λ (y) := u,λ (x + y) → w(y) in C 1 locally, where w is the unique solution of
(2.8),
(3) the associated critical value can be estimated as follows:
2d(x , ∂Ω) 1 + o(1)
.
(2.111)
c,λ = N I [w] + exp −
36
37
38
39
40
41
42
43
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Existence and stability of spikes
1
519
3. Stability and instability in the shadow system case
1
2
3
4
5
6
7
2
As we have already seen in Section 2 that there are many single and multiple spike solutions
for the shadow system (2.2). The question is: are they all stable with respect to the shadow
system (2.2)? Unfortunately, as we will show below, only one of them is stable.
Let u be a (boundary or interior) spike solution. Then it is easy to see that (a , ξ )
defined by the following
8
9
a = ξq/(p−1) u ,
10
ξ =
1
|Ω|
−(p−1)/(qr−(p−1)(s+1))
Ω
ur
dx
13
14
15
16
17
18
19
20
21
22
23
(3.1)
28
29
34
35
36
37
38
39
40
By using (3.1), it is easy to see that the eigenvalues of problem (3.2) in H 2 (Ω) × L∞ (Ω)
are the same as the eigenvalues of the following eigenvalue problem
r−1
qr
2
p−1
Ω u φ p
φ − φ + pu φ −
u = α φ,
r
s + 1 + τ α
Ω u
φ ∈ H 2 (Ω).
45
16
17
(3.3)
18
19
20
21
22
23
24
25
26
A simple argument [8] shows that
27
T HEOREM 3.1. Any multiple-spike solution is linearly unstable for the shadow system
(2.2).
28
29
30
31
Let
32
L (φ) = 2 φ − φ + pup−1
φ,
r−1
qr
Ω u φ p
u .
L (φ) = L (φ) −
s + 1 + τ λ Ω ur
33
34
(3.4)
Thus we can only concentrate on the study of stability for single-spike solutions. The
study of stability and instability of single spike solutions can be divided into two parts:
small eigenvalues and large eigenvalues.
35
36
37
38
39
40
41
3.1. Small eigenvalues for L
43
44
10
15
(3.2)
41
42
9
14
32
33
7
13
30
31
6
12
25
27
5
11
is a solution pair of the stationary problem to the shadow system (2.2).
In this section, we analyze the following linearized eigenvalue problem
⎧
p−1
p
⎪
⎨ 2 φ − φ + p a q φ − q aq+1
η = α φ , ∂φ
∂ν = 0 on ∂Ω,
ξ
ξ
⎪ r ar−1 φ
1+s
⎩
τ |Ω| Ω
ξ s dx − τ η = α η.
24
26
4
8
11
12
3
42
43
In [73], it was proved that single boundary spike must concentrate at a critical point of
the mean curvature function H (P ). On the other hand, at a non-degenerate critical point of
44
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J. Wei
H (P ), there is also a single boundary spike. Furthermore, in [76], it is proved that the single
boundary spike at a non-degenerate critical point of H (P ) is actually non-degenerate.
Next we study the eigenvalue estimates associated with the linearized operator at u :
p−1
L = 2 − 1 + pu . (Here the domain of L is H 2 (Ω).) We first note the following
result.
6
7
φ − φ + pw p−1 φ = μφ
in RN , φ ∈ H 1 RN
12
μ2 = · · · = μN +1 = 0,
μ1 > 0,
μN +2 < 0, . . . .
(3.6)
P ROOF. This follows from Theorem 2.12 of [42] and Lemma 4.2 of [58].
15
22
23
24
27
28
20
T HEOREM 3.3. (See [76].) For sufficiently small, the following eigenvalue problem
p−1
2 φ − φ + pu φ = τ φ in Ω,
(3.7)
∂φ
on ∂Ω
∂ν = 0
31
32
33
34
35
36
37
38
39
40
41
42
45
22
23
24
26
27
28
29
j
τ
2
30
→ η0 λ j ,
j = 1, . . . , N − 1,
(3.8)
+
1
(Here w (|z|) denotes the radial derivative of w with respect to |z|.)
j
Furthermore the eigenfunction corresponding to τ , j = 1, . . . , N − 1, is given by the
following:
φj =
31
32
where λ1 λ2 · · · λN −1 are the eigenvalues of the matrix Gb (P0 ) := (∂i ∂j H (P0 )),
and
2
N − 1 R+N (w (|z|)) zN dz
η0 =
> 0.
(3.9)
∂w 2
N +1
R N ( ∂z ) dz
43
44
21
25
μ1
admits exactly (N −1) eigenvalues τ1 τ2 · · · τN −1 in the interval [ μN+1
2 , 2 ], where
μ1 and μN +1 are given by Lemma 3.2.
j
Moreover, we have the following asymptotic behavior of τ :
29
30
17
19
25
26
16
18
The small eigenvalues for L were characterized completely in [76].
20
21
13
14
Moreover, the eigenfunction corresponding to μ1 is radial and of constant sign.
18
19
9
11
16
17
5
10
admits the following set of eigenvalues:
14
15
4
8
(3.5)
12
13
3
7
10
11
2
6
L EMMA 3.2. The following eigenvalue problem
8
9
1
N
−1
i=1
∂w,P
aij + o(1)
∂τi (P )
33
34
35
36
37
38
39
40
41
42
43
(3.10)
44
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Existence and stability of spikes
1
2
521
where P is the local maximum point of u , aj = (a1j , . . . , a(N −1)j )T is the eigenvector
corresponding to λj , namely
3
j = 1, . . . , N − 1.
(3.11)
5
7
8
9
10
12
dμP0 (z) = lim ε→0
13
17
18
19
20
e−
∂Ω
14
16
2|z−P0 |
ε
e
23
2|z−P |
− ε 0
(3.12)
.
dz
∂Ω
28
29
30
31
32
35
36
37
38
39
40
41
42
∂Ω
ez−P0 ,a (z − P0 )i (z − P0 )j dμP0 (z) := Gi (P0 )
45
10
12
13
15
16
17
18
19
20
23
is non-singular. (H2)
24
25
Such a vector a is unique. Moreover, Gi (P0 ) is a positive definite matrix. A geometric
characterization of a non-degenerate peak point P0 is the following:
P0 ∈ interior (convex hull of support dμP0 (z) .
For a proof of the above facts, see Theorem 5.1 of [74].
In [75] and [74], the author proved the following theorem.
26
27
28
29
30
31
32
33
T HEOREM 3.4. Suppose that P0 is a non-degenerate peak point. Then for 1, there
exists a single interior spike solution u concentrating at P0 . Furthermore, u is locally
unique. Namely, if there are two families of single interior spike solutions u,1 and u,2 of
(2.4) such that P1 → P0 , P2 → P0 where
u,1 P1
= max u (P ),
P ∈Ω̄
u,2 P2
= max u,2 (P ),
P ∈Ω̄
then P1 = P2 , u,1 = u,2 . Moreover,
43
44
9
22
33
34
8
21
and
25
27
7
14
It is easy to see that the support of dμP0 (z) is contained in B̄d(P0 ,∂Ω) (P0 ) ∩ ∂Ω.
A point P0 is called “non-degenerate peak point” if the followings hold: there exists
a ∈ RN such that
ez−P0 ,a (z − P0 ) dμP0 (z) = 0
(H1)
24
26
6
11
dz
21
22
4
5
For single interior spikes, we obtain similar results. But it becomes more involved since
now the error is exponentially small.
The existence of interior spike solutions depends highly on the geometry of the domain.
In [74] and [75], the author first constructed a single interior spike solution. To state the
result, we need to introduce some notations. Let
11
15
2
3
Gb (P0 )
aj = λj aj ,
4
6
1
P1
= P2
1
= P0 + d(P0 , ∂Ω)a + o(1)
as → 0.
2
34
35
36
37
38
39
40
41
42
43
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2
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J. Wei
Let w,P and ϕ,P be defined as in Section 2.3. (It was proved in [75] and [74] that
− log[−ϕ,P (P )] → 2d(P , ∂Ω) as → 0.)
Similarly, we obtain the following eigenvalue estimates for u
4
5
8
6
φ = τφ
2 φ − φ + pup−1
∂φ
= 0 on ∂Ω
∂ν
in Ω,
7
(3.13)
13
14
15
16
19
τj = c0 + o(1) ϕ,P0 (P0 )λj −1 ,
τl = μl + o(1),
l N + 2,
22
23
24
25
26
27
c0 = 2d
30
31
32
33
34
35
12
13
14
15
−2
(P0 , ∂Ω)
RN
pw p−1 w u∗ (r)
∂w 2
RN ( ∂y1 ) dy
16
17
< 0,
(3.14)
u − u = 0,
u(0) = 1,
u = u(r)
in R .
N
N
∂w,P =
,
aj −1,l + o(1) ∂Pl P =P
22
(3.15)
where aj = (aj,1 , . . . , aj,N
Gi (P0 )
aj = λj aj ,
(3.16)
is the eigenvector corresponding to λj , namely
j = 1, . . . , N.
36
41
44
45
26
27
29
30
31
32
33
34
35
38
39
Let α be an eigenvalue of (3.3). Then the following holds. (The proof of it is routine. See
Appendix of [77].)
42
43
25
37
3.2. A reduction lemma
39
40
24
36
37
38
23
28
l=1
)t
19
21
Furthermore, the eigenfunction (suitably normalized) corresponding to τj , j = 2, . . . ,
N + 1, is given by the following:
φj
18
20
where u∗ (r) is the unique radial solution of the following problem
28
29
11
j = 2, . . . , N + 1,
where λj , j = 1, . . . , N , are the eigenvalues of Gi (P0 ) and
20
21
10
τ1 = μ1 + o(1),
17
18
8
9
admits the following set of eigenvalues:
11
12
3
5
T HEOREM 3.5. The following eigenvalue problem
9
10
2
4
6
7
1
40
41
42
L EMMA A.
(1) α = o(1) if and only if α = (1 + o(1))τj for some j = 2, . . . , N + 1, where τj is
given by Theorem 3.3 or Theorem 3.5.
43
44
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Existence and stability of spikes
1
523
(2) If α → α0 = 0. Then α0 is an eigenvalue of the following eigenvalue problem
2
qr
φ − φ + pw p−1 φ −
s + 1 + τ α0
φ ∈ H 2 RN .
3
4
5
6
2
w r−1 φ p
w = α0 φ,
r
RN w
3
RN
4
5
(3.17)
7
8
8
9
C OROLLARY 3.6. For 1, (a , ξ ) is unstable with respect to the shadow system (2.2).
11
12
3.3. Large eigenvalues: NLEP method
13
14
15
16
17
18
19
20
21
22
14
This section is devoted to the study of the non-local eigenvalue problem (3.17). By [77]
and [78], if problem (3.17) admits an eigenvalue λ with positive real part, then all single point-condensation solutions are unstable, while if all eigenvalues of problem (3.17)
have negative real part, then all single point-condensation solutions are either stable or
metastable. (Here we say that a solution is metastable if the eigenvalues of the associated linearized operator either are exponentially small or have strictly negative real parts.)
Therefore it is vital to study problem (3.17).
We first consider the simple case when τ = 0. Namely, we study the following NLEP:
23
24
φ − φ + pw
25
p−1
φ − γ (p − 1)
26
27
30
31
w r−1 φ p
w = λφ,
r
RN w
RN
φ ∈ H RN ,
2
(3.18)
34
35
36
37
38
39
40
41
γ :=
18
19
20
21
22
24
25
28
qr
,
(s + 1)(p − 1)
λ ∈ C,
λ = 0,
φ(x) = φ |x| .
29
30
(3.19)
31
32
For problem (3.18), it is known that when γ = 0, there exists an eigenvalue λ = μ1 > 0
(Lemma 3.2). An important property of (3.18) is that non-local term can push the eigenvalues of problem (3.18) to become negative so that the point-condensation solutions of the
Gierer–Meinhardt system become stable or metastable.
A major difficulty in studying problem (3.18) is that the left-hand side operator is not
self-adjoint if r = p + 1. (In the classical Gierer–Meinhardt system, r = 2, p = 2.) Therefore it may have complex eigenvalues or Hopf bifurcations. Many traditional techniques
do not work here.
In [78] and [77], the eigenvalues of problem (3.18) in the following two cases
33
34
35
36
37
38
39
40
41
42
r = 2,
or r = p + 1
44
45
17
27
42
43
16
26
32
33
15
23
where
28
29
10
11
12
13
6
7
A direct application of Theorem 3.5 is the following corollary.
9
10
1
43
44
are studied and the following results are proved.
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1
2
J. Wei
T HEOREM 3.7.
(1) If (p, q, r, s) satisfies
3
4
(A)
5
6
7
1
2
3
qr
γ=
> 1,
(s + 1)(p − 1)
4
5
6
and
8
9
F:chipot506.tex; VTEX/Rita p. 38
(B)
r = 2,
1<p1+
4
N
or r = p + 1, 1 < p <
10
11
12
13
14
15
16
8
,
+2
N +2
=N
N −2 when N 3 and ( N −2 )+ = +∞ when N = 1, 2.
Re(λ) < −c1 < 0 for some c1 > 0, where λ = 0 is an eigenvalue of
where
Then
(3.18).
(2) If γ < 1, problem (3.18) has an eigenvalue λ1 > 0.
(3) If
(C)
r = 2,
4
p>1+
N
9
+
+2
(N
N −2 )+
17
18
N +2
N −2
7
10
11
problem
15
16
17
and 1 < γ < 1 + c0 ,
18
19
for some c0 > 0. Then problem (3.18) has an eigenvalue λ1 > 0.
20
21
22
23
24
25
21
We give a complete proof of Theorem 3.7 since this is the key element in all the stability
result later on.
The proof of Theorem 3.7 is based on the following important inequalities which are
new and interesting.
26
27
28
29
30
31
32
33
34
39
40
41
42
43
44
45
23
24
25
27
28
2(p − 1) RN wφ RN w p φ
2
2
p−1 2
|∇φ| + φ − pw
φ +
2
RN
RN w
2
p+1 RN w
−(p − 1)
wφ
( RN w 2 )2 RN
a1 dL2 2 (RN ) (φ, X1 ),
36
38
22
26
L EMMA 3.8. Let w be the unique solution to (2.8).
(1) If 1 < p < 1 + N4 , then there exists a positive constant a1 > 0 such that
35
37
13
14
19
20
12
for all φ ∈ H 1 (RN ), where X1 := span{w,
(2) If p = 1 +
4
N,
29
30
31
32
33
34
(3.20)
∂w
∂yj ,
j = 1, . . . , N }.
35
36
37
38
then there exists a positive constant a2 > 0 such that
39
2(p − 1) RN wφ RN w p φ
2
2
p−1 2
|∇φ| + φ − pw
φ +
2
RN
RN w
2
p+1 N w
− (p − 1) R
wφ
( RN w 2 )2 RN
40
41
42
43
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Existence and stability of spikes
525
a2 dL2 2 (RN ) (φ, X2 ),
1
(3.21)
2
3
4
5
2
1
1
∂w
p−1 w + 2 y∇w(y), ∂yj ,
for all φ
2 := span{w,
(3) There exists a positive constant a3 > 0 such that
∈ H 1 (RN ), where X
j = 1, . . . , N}.
5
(p − 1)( RN w p φ)2
|∇φ|2 + φ 2 − pw p−1 φ 2 +
p+1
RN
RN w
a3 dL2 2 (RN ) (φ, X1 ), ∀φ ∈ H 1 RN .
7
8
9
6
7
8
(3.22)
10
12
13
14
15
18
and L0 := −1+pw p−1 . Note that L is not selfadjoint if r = p +1.
21
L0 w = (p − 1)w p ,
L0
23
1
1
w + x∇w = w
p−1
2
(3.23)
28
RN
−1 L0 w w =
RN
w
1
1
w + x∇w =
p−1
2
1
N
−
p−1
4
32
RN
−1 p
L0 w w =
39
40
41
42
43
44
45
=
36
38
25
27
31
37
24
26
and
30
35
17
21
28
34
15
20
26
33
14
22
25
29
13
19
Then
24
27
12
18
∂w X0 := kernel(L0 ) = span
j = 1, . . . , N .
∂yj
20
23
11
16
qr
(p−1)(s+1)
where γ =
Let
19
22
9
10
P ROOF OF L EMMA 3.8. To this end, we first introduce some notations and make some
preparations. Set
r−1 φ
N w
Lφ := L0 φ − γ (p − 1) R
w p , φ ∈ H 2 RN
r
RN w
16
17
3
4
6
11
1
RN
RN
w
p
1
1
w + x∇w
p−1
2
−1 L0 w
RN
29
w2 ,
30
(3.24)
32
33
1
1
L0 w =
p−1
p−1
31
34
35
36
2
RN
w .
Since L is not selfadjoint, we introduce a new operator as follows:
p
N wφ
N w φ
p
R
R
L1 φ := L0 φ − (p − 1) w
−
(p
−
1)
w
2
2
RN w
RN w
p+1
RN w
RN wφ
w.
+ (p − 1)
2
( RN w )2
(3.25)
37
38
39
40
41
42
43
(3.26)
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J. Wei
By (3.26), L1 is selfadjoint. Next we compute the kernel of L1 . It is easy to see that
∂w
w, ∂y
, j = 1, . . . , N, ∈ kernel(L1 ). On the other hand, if φ ∈ kernel(L1 ), then by (3.23)
j
3
5
6
7
9
12
13
14
15
19
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
7
8
9
10
11
12
13
14
15
16
φ − c1 (φ)
18
22
6
Hence
17
21
5
p
p+1
N w φ
N w
R
R
RN wφ
c1 (φ) = (p − 1) − (p − 1)
,
2
2
( R N w )2
RN w
N wφ
c2 (φ) = (p − 1) R
.
2
RN w
11
20
4
where
10
16
2
3
L0 φ = c1 (φ)w + c2 (φ)w p
1
w
1
w + x∇w + c2 (φ)L0
= c1 (φ)L0
p−1
2
p−1
4
8
1
1
1
1
w + x∇w − c2 (φ)
w ∈ kernel(L0 ).
p−1
2
p−1
17
(3.27)
18
19
20
Note that
21
1
w p ( p−1
w + 12 x∇w)
c1 (φ) = (p − 1)c1 (φ)
2
RN w
1
1
p+1
RN w
RN w( p−1 w + 2 x∇w)
− (p − 1)c1 (φ)
( RN w 2 )2
N RN w p+1
1
−
= c1 (φ) − c1 (φ)
2
p−1
4
RN w
RN
by (3.24) and (3.25). This implies that c1 (φ) = 0. By (3.27) and Lemma 3.2, this shows
that the kernel of L1 is exactly X1 .
Now we prove (3.20). Suppose (3.20) is not true, then there exists (α, φ) such that (i) α
∂w
is real and positive, (ii) φ ⊥ w, φ ⊥ ∂y
, j = 1, . . . , N , and (iii) L1 φ = αφ.
j
We show that this is impossible. From (ii) and (iii), we have
p
N w φ
(L0 − α)φ = (p − 1) R
w.
(3.28)
2
RN w
We first claim that RN w p φ = 0. In fact if RN w p φ = 0, then α > 0 is an eigenvalue of
L0 . By Lemma 3.2, α = μ1 and φ has constant sign. This contradicts with the fact that
φ ⊥ w. Therefore α = μ1 , 0, and hence L0 − α is invertible in X0⊥ . So (3.28) implies
p
N w φ
(L0 − α)−1 w.
φ = (p − 1) R
2
w
N
R
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
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Existence and stability of spikes
1
2
3
4
5
6
7
8
9
Thus
1
p N w φ
p
w φ = (p − 1) R
(L0 − α)−1 w w p ,
2
RN
RN
RN w
(L0 − α)−1 w w p ,
w 2 = (p − 1)
RN
0=
12
13
14
15
16
w =
2
RN
Let h1 (α) =
RN
19
3
4
5
6
7
(L0 − α)−1 w (L0 − α)w + αw ,
(L0 − α)−1 w w.
8
9
10
(3.29)
RN
=
−1 L0 w w =
N
1
−
p−1
4
13
then
RN
1
1
w + x · ∇w w
p−1
2
RN
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
15
16
17
w >0
19
20
since 1 < p < 1 +
22
23
14
18
2
20
21
11
12
− α)−1 w)w,
17
18
RN
RN ((L0
h1 (0) =
2
RN
10
11
527
h1 (α) =
4
N.
Moreover
RN
−2 L0 − α w w =
21
RN
2
(L0 − α)−1 w > 0.
22
23
24
This implies h1 (α) > 0 for all α ∈ (0, μ1 ). Clearly, also h1 (α) < 0 for α ∈ (μ1 , ∞) (since
limα→+∞ h1 (α) = 0). This is a contradiction to (3.29)!
This proves the inequality (3.20).
The proof of (3.21) is similar. In this case we have
−1 1
1
w + x∇w = 0.
(3.30)
L0 w w =
w
p−1
2
RN
RN
25
Thus the kernel of L1 is X2 . The rest of the proof is exactly the same as before.
To prove (3.22), we introduce
p
N w φ
L3 φ := L0 φ − (p − 1) R p+1 w p .
(3.31)
RN w
33
Similar as before, the kernel of L3 is exactly X1 .
Suppose (3.22) is not true, then there exists (α, φ) such that (a) α is real and positive,
∂w
(b) φ ⊥ w, φ ⊥ ∂y
, j = 1, . . . , N , and (c) L3 φ = αφ.
j
We show that this is impossible. From (a) and (c), we have
(p − 1) RN w p φ p
(L0 − α)φ =
w .
(3.32)
p+1
RN w
39
26
27
28
29
30
31
32
34
35
36
37
38
40
41
42
43
44
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1
2
3
J. Wei
Similar to the proof of (3.20), we have that
invertible in X0⊥ . So (3.32) implies
4
φ=
5
6
7
8
F:chipot506.tex; VTEX/Rita p. 42
(p − 1)
RN
wp φ
RN
w p+1
w p φ = 0, α = μ1 , 0, and hence L0 − α is
RN
2
3
4
(L0 − α)−1 w p .
5
6
7
Thus
8
p N w φ
p
(L0 − α)−1 w p w p ,
w φ = (p − 1) R p+1
RN
RN
RN w
(L0 − α)−1 w p w p .
w p+1 = (p − 1)
9
10
11
12
RN
13
9
10
11
(3.33)
RN
14
15
h3 (α) = (p − 1)
RN
18
h3 (0) = (p − 1)
23
28
29
30
17
18
RN
p
p
(L−1
0 w )w −
20
RN
21
w p+1 = 0.
22
23
Moreover
24
25
27
RN
16
w p+1 ,
19
22
26
(L0 − α)−1 w p w p −
then
21
24
h3 (α) = (p − 1)
RN
(L0 − α)−2 w p w p = (p − 1)
RN
2
(L0 − α)−1 w p > 0.
This implies h3 (α) > 0 for all α ∈ (0, μ1 ). Clearly, also h3 (α) < 0 for α ∈ (μ1 , ∞). A contradiction to (3.33)!
31
32
35
36
37
38
39
40
41
42
43
44
45
25
26
27
28
29
30
31
Using Lemma 3.8, we can prove Theorem 3.7(i).
32
33
34
13
15
17
20
12
14
Let
16
19
1
33
P ROOF OF T HEOREM 3.7( I ). We divide the proof into three cases.
C ASE 1. r = 2, 1 < p < 1 +
34
35
4
N.
36
Let α0 = αR + iαI and φ = φR + iφI . Since α0 = 0, we can choose φ ⊥ kernel(L0 ).
Then we obtain two equations
N wφR
L0 φR − (p − 1)γ R
w p = αR φR − αI φI ,
2
RN w
N wφI
L0 φI − (p − 1)γ R
w p = αR φI + αI φR .
2
w
N
R
37
38
39
40
(3.34)
41
42
43
(3.35)
44
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Existence and stability of spikes
1
2
3
4
5
6
7
8
9
10
11
529
Multiplying (3.34) by φR and (3.35) by φI and adding them together, we obtain
2
φR + φI2
−αR
1
2
3
4
RN
= L1 (φR , φR ) + L1 (φI , φI )
p
p
R + RN wφI RN w φI
RN wφR RN w φ
+ (p − 1)(γ − 2)
2
RN w
2 2 p+1 RN w
.
+ (p − 1)
wφR +
wφI
( RN w 2 )2
RN
RN
5
6
7
8
9
10
11
12
13
14
15
16
17
18
12
Multiplying (3.34) by w and (3.35) by w we obtain
N wφR
p
R
(p − 1)
w φR − γ (p − 1) w p+1
2
RN
RN
RN w
= αR
wφR − αI
wφI ,
RN
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
36
37
38
39
40
41
42
43
44
45
14
15
16
17
(3.36)
RN
20
N wφI
p
R
(p − 1)
w φI − γ (p − 1) w p+1
2
RN
RN
RN w
= αR
wφI + αI
wφR .
RN
21
22
23
(3.37)
RN
Multiplying (3.36) by RN wφR and (3.37) by RN wφI and adding them together, we obtain
p
wφR
w φR + (p − 1)
wφI
w p φI
(p − 1)
RN
RN
= αR + γ (p − 1) R
N
w p+1
RN
18
19
34
35
13
w2
RN
RN
wφR
2
RN
+
RN
2 wφI
24
25
26
27
28
29
30
31
32
.
Therefore we have
2
−αR
φR + φI2
RN
= L1 (φR , φR ) + L1 (φI , φI )
p+1 (
2
2
1
N w
N wφR ) + ( RN wφI )
R
R
+ (p − 1)(γ − 2)
αR + γ 2
2
p−1
RN w
RN w
2 2 p+1 N w
.
+ (p − 1) R
wφ
+
wφ
R
I
( RN w 2 )2
RN
RN
33
34
35
36
37
38
39
40
41
42
43
44
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1
F:chipot506.tex; VTEX/Rita p. 44
J. Wei
Set
1
2
2
φR = cR w + φR⊥ , φR⊥ ⊥ X1 ,
3
φI = cI w + φI⊥ ,
φI⊥ ⊥ X1 .
3
4
5
4
5
Then
6
7
8
RN
wφR = cR
RN
w2 ,
RN
2
dL2 2 (RN ) (φR , X1 ) = φR⊥ L2 ,
9
10
6
wφI = cI
RN
7
w2 ,
8
2
dL2 2 (RN ) (φI , X1 ) = φI⊥ L2 .
9
10
11
12
11
By some simple computations we have
12
13
13
L1 (φR , φR ) + L1 (φI , φI )
2
2
+ (γ − 1)αR cR + cI
14
15
16
RN
17
2
w 2 + (p − 1)(γ − 1)2 cR
+ cI2
2 2
+ αR φR⊥ L2 + φI⊥ L2 = 0.
18
14
15
RN
w p+1
16
17
18
19
20
19
20
By Lemma 3.8 (1)
21
2
(γ − 1)αR cR
+ cI2
22
23
24
26
28
31
32
33
34
37
38
39
40
41
42
43
44
45
23
24
RN
w p+1
25
26
27
28
29
Since γ > 1, we must have αR < 0, which proves Theorem 3.7 in Case 1.
C ASE 2. r = 2, p = 1 +
30
31
4
N.
32
33
Set
34
35
36
22
w2
2 2
+ (αR + a1 ) φR⊥ L2 + φI⊥ L2 0.
27
30
RN
2
+ (p − 1)(γ − 1)2 cR
+ cI2
25
29
21
w0 =
35
1
1
w + x∇w.
p−1
2
(3.38)
37
We just need to take care of w0 .
Suppose that α0 = 0 is an eigenvalue of L. Let α0 = αR + iαI and φ = φR + iφI . Since
α0 = 0, we can choose φ ⊥ kernel(L0 ). Then similar to Case 1, we obtain two equations
(3.34) and (3.35). We now decompose
φR = cR w + bR w0 + φR⊥ ,
φI = cI w + bI w0 + φI⊥ ,
36
φR⊥ ⊥ X1 ,
φI⊥
⊥ X1 .
38
39
40
41
42
43
44
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Existence and stability of spikes
1
531
Similar to Case 1, we obtain
1
2
3
4
5
6
7
8
2
L1 (φR , φR ) + L1 (φI , φI )
2
+ (γ − 1)αR cR
+ cI2
2
+ αR bR
RN
9
10
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
RN
2
2
w 2 + (p − 1)(γ − 1)2 cR
+ cI2
+ bI2
w02
RN
2
w02
RN
4
w p+1
5
6
2
2
+ φR⊥ L2 + φI⊥ L2 0
7
8
9
10
By Lemma 3.8(2)
11
12
3
2
(γ − 1)αR cR
+ cI2
2
+ αR bR
RN
RN
2
w 2 + (p − 1)(γ − 1)2 cR
+ cI2
2
w02
+ bI2
RN
2 w02
11
RN
12
w p+1
13
2 2
+ (αR + a2 ) φR⊥ L2 + φI⊥ L2
33
34
35
36
37
38
39
40
41
42
43
44
45
15
16
0.
17
If αR 0, then necessarily we have
19
cR = cI = 0,
18
φR⊥
= 0,
20
φI⊥
21
= 0.
22
23
Hence φR = bR w0 , φI = bI w0 . This implies that
bR L0 w0 = (bR − bI )w0 ,
24
25
bI L0 w0 = (bR + bI )w0 ,
26
27
which is impossible unless bR = bI = 0. A contradiction!
28
29
+2
C ASE 3. r = p + 1, 1 < p < ( N
N −2 )+ .
30
31
32
14
31
Let r = p + 1. L becomes
32
p
qr
N w ·
R
L = L0 −
wp .
p+1
s + 1 RN w
33
34
35
36
We will follow the proof of Case 1.
Let α0 = αR + iαI and φ = φR + iφI . Since α0 = 0, we can choose φ ⊥ kernel(L0 ).
Then similarly we obtain two equations
p
N w φR
L0 φR − (p − 1)γ R
w p = αR φR − αI φI ,
p+1
w
N
R
p
N w φI
L0 φI − (p − 1)γ R
w p = αR φI + αI φR .
p+1
w
N
R
37
38
39
40
41
(3.39)
42
43
(3.40)
44
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2
3
4
J. Wei
Multiplying (3.39) by φR and (3.40) by φI and adding them together, we obtain
2
−αR
φR + φI2 = L3 (φR , φR ) + L3 (φI , φI )
RN
6
7
9
10
11
12
13
14
15
16
17
1
2
3
4
( RN w p φR )2 + ( RN w p φI )2
.
+ (p − 1)(γ − 1)
p+1
RN w
5
8
F:chipot506.tex; VTEX/Rita p. 46
5
6
7
8
By Lemma 3.8(3)
2
αR
φR + φI2 + a3 dL2 2 (φ, X1 )
9
10
11
RN
( N w p φR )2 + ( RN w p φI )2
0
+ (p − 1)(γ − 1) R
p+1
RN w
12
13
14
15
which implies αR < 0 since γ > 1.
Theorem 3.7(i) in Case 3 is thus proved.
16
18
19
20
21
22
18
P ROOF OF T HEOREM 3.7( II ). Assume that γ < 1. To prove Theorem 3.7(ii), we introduce
the following function:
h4 (λ) :=
(L0 − λ)−1 w p w r−1 .
w r − γ (p − 1)
(3.41)
RN
23
24
25
26
27
28
29
30
31
32
RN
Note that h4 (λ) is well defined in (0, μ1 ), where μ1 is the unique positive eigenvalue of
L0 . Let us denote the corresponding eigenfunction by Φ0 . Since μ1 is a principal eigenvalue, we may assume that Φ0 > 0.
It is easy to see that to prove Theorem 3.7(ii), it is enough to find a positive zero of
h4 (λ).
First we have
p r−1
h4 (0) =
w r − γ (p − 1)
L−1
w
w
=
(1
−
γ
)
w r > 0.
(3.42)
0
RN
RN
RN
39
40
43
44
45
22
24
25
26
27
28
29
30
31
32
34
(3.43)
36
37
Multiplying (3.43) by Φ0 and integrating by parts, we have
(μ1 − λ)
Φλ Φ0 =
Φ0 w p ,
RN
41
42
21
35
(L0 − λ)Φλ = w p .
37
38
20
33
Set Φλ = (L0 − λ)−1 w p . Then Φλ satisfies
35
36
19
23
33
34
17
which implies that
Φλ Φ0 =
RN
RN
38
39
40
41
42
1
μ1 − λ
43
RN
Φ0 w p .
44
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Existence and stability of spikes
1
Let
1
2
3
4
533
Φλ =
1
(μ1 − λ) RN Φ02
Φ0 w Φ0 + Φλ⊥ ,
p
RN
2
Φλ⊥
⊥ Φ0 .
(3.44)
4
5
6
7
8
9
10
11
5
Then as λ → μ1 , λ < μ1 , we have that Φλ⊥ L2 (RN ) is uniformly bounded and by (3.44)
RN
8
Φλ w r−1 → +∞,
9
10
which implies that
11
12
h4 (λ) → −∞ as λ → μ1 , λ < μ1 .
(3.45)
14
15
16
19
20
21
15
RN
RN
18
19
(3.46)
22
h5 (λ) = (1 − γ )
27
= (1 − γ )
28
30
31
23
RN
w 2 − γ (p − 1)λ
RN
26
29
w − γ (p − 1)λ
2
RN
RN
w (L0 − λ)−1 (w)
24
25
26
2
wL−1
0 (w) + O λ .
27
28
Since RN wL−1
0 (w) < 0, we see that for 0 < γ − 1 small, there is a small λ0 > 0 such
that h5 (λ0 ) > 0.
32
33
36
39
40
41
42
45
31
34
35
36
D(r) :=
(p − 1)
−1 r−1 r−1 w
RN L0 w
RN
( RN w r )2
where L0 = − 1 +
u(|x|)}) and
pw p−1 (L−1
0
37
w2
>0
(3.47)
38
39
40
exists in Hr2 (RN ) = {u ∈ H 2 (RN ) | u(x) =
43
44
30
33
T HEOREM 3.9.
(1) If
37
38
29
32
For general r, the author in [80] proved the following:
34
35
20
21
We write it as
24
25
16
17
P ROOF OF T HEOREM 3.7( III ). Similarly, we just need to find a zero of
2
w − γ (p − 1)
w(L0 − λ)−1 w p .
h5 (λ) :=
22
23
13
14
By (3.42) and (3.45), there is a λ0 ∈ (0, μ1 ) such that h4 (λ0 ) = 0.
This proves (ii) of Theorem 3.7.
17
18
6
7
12
13
3
41
42
43
1
1
1+ √
<γ <1+ √
,
1 + ρ0
1 − ρ0
44
(3.48)
45
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1
2
3
4
5
6
7
8
9
10
11
12
13
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J. Wei
where ρ0 > 0 is given by
p+1
N w
ρ0 := ) R
< 1.
2p
2
RN w
RN w
1
2
(3.49)
16
17
Then for any non-zero eigenvalue λ of problem (3.18), we have Re(λ) < −c1 < 0
for some c1 > 0.
(2) If (p, q, r, s) satisfies
2r
N +2
1+
<p<
and γ < 1 + c0 ,
(3.50)
N
N −2 +
for some c0 > 0. Then problem (3.18) has a real eigenvalue λ1 > 0.
24
25
26
29
30
31
32
33
F (r) = 1 −
36
37
40
41
42
43
12
15
16
17
22
2r
N
23
and
|γ − 2| *
γ −2
F (p + 1) +
F (p + 1)(F (p + 1) − F (2)),
F (r) γ
γ
24
25
(3.51)
26
27
then for any non-zero eigenvalue λ of problem (3.18), we have Re(λ) < −c1 < 0 for some
c1 > 0.
R EMARK . Condition (3.51) holds if 2 < r < p + 1, F (r) 0 (i.e., 1 < p 1 + 2r
N ) and
1 < γ 2. Thus in this case we obtain the stability of the non-zero eigenvalues of (3.18).
This is the first explicit result for the case when r ∈
/ {2, p + 1}. For γ > 2, we need
F (r) *
γ − 2
F (p + 1) − F (p + 1)(F (p + 1) − F (2)) .
γ
Going back to the shadow system case, the following result was proved in [77].
28
29
30
31
32
33
34
35
36
37
38
T HEOREM 3.11. Assume that 1 and τ is small. If (p, q, r, s) satisfy (A) and (B) in
Theorem 3.7, then
(1) single boundary spike solution at a non-degenerate local maximum point of mean
curvature is stable, and
(2) single interior spike solution is metastable.
44
45
11
21
38
39
10
20
p−1
N.
2r
Suppose 2 < r < p + 1, 1 < p < 1 +
34
35
9
19
27
28
8
18
T HEOREM 3.10. Let
22
23
7
14
Generally speaking, D(r) is very difficult to compute. A recent result of the author and
L. Zhang partially solved this problem and moreover we obtained more general and explicit
result. For example the following result are proved [81].
20
21
6
13
18
19
4
5
14
15
3
39
40
41
42
43
44
Related work can also be found in [59] and [60].
45
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Existence and stability of spikes
1
535
3.4. Uniqueness of Hopf bifurcations
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
2
In Section 3.3, we have discussed the NLEP (3.17) when τ = 0. It is easy to see that when
τ small, results in Theorem 3.7 still hold. On the other hand, for τ large, it is easy to see
that there is an unstable eigenvalue [8] to (3.17). (In fact, as τ → +∞, there is a positive
eigenvalue near μ1 > 0.) Therefore, as τ varies from 0 to ∞, Hopf bifurcation may occur.
In this section, we show that in some special cases, Hopf bifurcation is actually unique.
We consider the following non-local eigenvalue problem (putting r = p = 2, s = 0 in
(3.17))
γ
N wφ
R
Lφ := φ − φ + 2wφ −
w 2 = λ0 φ,
1 + τ λ0 R 2 w 2
21
22
23
24
25
26
27
28
φ ∈ H 2 RN .
33
34
35
36
37
N wφ
w2 = 0
L0 φ − γ R
2
w
N
R
which implies that
L0
N wφ
φ − γ R
w
=0
2
RN w
and hence by Lemma 3.2
N wφ
φ − γ R
w ∈ X0 .
2
RN w
38
39
40
41
42
45
7
8
9
12
13
14
15
16
17
18
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
This is impossible since φ is radially symmetric and φ = cw for all c ∈ R.
Thus there must be a point
√ τ1 at which L has a Hopf bifurcation, i.e., L has a purely
imaginary eigenvalue α = −1αI . To prove Theorem 3.12, all we need to show is that τ1
is unique. That is
43
44
6
19
P ROOF OF T HEOREM 3.12. Let γ > 1. As in [8], we may consider radially symmetric
functions only. By Theorem 1.4 of [77], for τ = 0 (and by perturbation, for τ small), all
eigenvalues lie on the left half plane. By [8], for τ large, there exist unstable eigenvalues.
Note that the eigenvalues will not cross through zero: in fact, if λ0 = 0, then we have
30
32
5
11
(3.52)
T HEOREM 3.12. Let L be defined by (3.52). Assume that N 3 and γ > 1. Then there
exists a unique τ = τ1 > 0 such that for τ < τ1 , (3.52) admits a positive eigenvalue, and
for τ > τ1 , all non-zero eigenvalues of problem (3.52) satisfy Re(λ) < 0. At τ = τ1 , L has
a Hopf bifurcation.
29
31
4
10
19
20
3
39
40
41
42
43
L EMMA 3.13. Let γ > 1. Then there exists a unique τ1 > 0 such that L has a Hopf bifurcation.
44
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1
2
3
4
5
6
7
8
Let φ0 = φ0R +
√
−1φ0I . Then from (3.53), we obtain the two equations
11
12
13
14
15
(3.55)
28
29
30
31
32
33
40
41
42
45
14
19
21
−1
φ0R = L0 L20 + αI2 w 2 .
22
(3.56)
23
24
Substituting (3.56) into (3.54) and (3.55), we obtain
2
2 −1 2
1
RN [wL0(L0 + αI ) w ]
= ,
2
γ
RN w
2 + α 2 )−1 w 2 ]
τ
I
RN [w(L
0
= .
2
γ
w
R2
25
26
(3.57)
27
28
29
(3.58)
30
31
32
Let
33
h6 (αI ) =
(L20
RN wL0
R2
+ αI2 )−1 w 2
.
w2
34
35
36
Then integration by parts gives
2 −1 w 2
2 2
N w (L0 + αI )
h6 (αI ) = R
.
2
RN w
37
Note that
42
43
44
13
20
36
39
11
17
L0 φ0I = αI φ0R .
−1
φ0I = αI L20 + αI2 w 2 ,
35
38
7
16
So φ0R = αI−1 L0 φ0I and
34
37
6
15
24
27
5
18
23
26
4
12
L0 φ0R = w 2 − αI φ0I ,
22
25
3
10
(3.54)
Note that (3.54) is independent
of τ .
Let us now compute RN wφ0R . Observe that (φ0R , φ0I ) satisfies
19
21
2
8
18
20
1
9
R
1
N wφ0
R
= ,
N
γ
R2 w
I
τ αI
N wφ0
R
.
=
N
γ
w
2
R
10
17
J. Wei
√
P ROOF. Let λ0 = −1αI be an eigenvalue
of L. Without loss of generality, we may
√
assume
that
α
>
0.
(Note
that
−
−1α
is
also
an eigenvalue of L.) Let φ0 = (L0 −
I
I
√
−1αI )−1 w 2 . Then (3.52) becomes
√
1 + τ −1αI
N wφ0
R
.
(3.53)
=
2
γ
RN w
9
16
F:chipot506.tex; VTEX/Rita p. 50
h6 (αI ) = −2αI
38
39
40
41
43
2 2 + α 2 )−2 w 2
I
RN w (L
0
< 0.
2
w
N
R
44
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Existence and stability of spikes
1
So since
1
2
3
h6 (0) =
4
RN
5
6
7
8
537
2
2
w(L−1
0 w )
> 0,
2
RN w
3
4
5
h6 (αI ) → 0 as αI → ∞, and γ > 1, there exists a unique αI > 0 such that (3.57) holds.
Substituting this unique αI into (3.58), we obtain a unique τ = τ1 > 0.
Lemma 3.13 is thus proved.
9
10
17
18
19
13
14
In all the previous sections, it is always assumed that is small. However, in practical applications, it is vital to know how small should be. The finite case has been completely
characterized in one-dimensional case by Wei and Winter [89]. We summarize the results
here.
Without loss of generality, we may assume that Ω = (0, 1). That is, we consider
⎧
ap
2
⎪
⎨ at = axx − a + ξ q , 0 < x < 1, t > 0,
1
τ ξt = −ξ + ξ −s 0 a r dx,
⎪
⎩
a > 0, ax (0, t) = ax (1, t) = 0.
20
21
22
23
24
25
26
27
29
ξ
qr
1+s− p−1
44
45
27
30
2
33
p
34
ux (0) = ux (1) = 0.
(3.60)
35
36
Letting
37
38
43
26
29
36
42
25
32
ux (x) < 0, 0 < x < 1,
41
22
31
35
40
19
28
uxx − u + u = 0,
39
18
24
ur (x) dx
34
37
17
23
and
33
16
21
(3.59)
0
30
32
1
=
15
20
The steady-state problem of (3.59) is equivalent to the following problem for the trans− q
formed function u given by u (x) = ξ p−1 a(x):
28
31
10
12
3.5. Finite case
14
16
8
11
12
15
7
9
Theorem 3.12 now follows from Lemma 3.13.
11
13
6
38
1
L :=
(3.61)
40
and rescaling u(x) = wL (y), where y = Lx, we see that wL satisfies the following ODE:
p
0 < y < L,
41
42
43
wL − wL + wL = 0,
wL (y) < 0,
39
44
wL (0) = wL (L) = 0.
(3.62)
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J. Wei
Since (3.62) is an autonomous ODE, it is easy to see that a non-trivial solution exists if
and only if
√
p−1
<
π
3
4
5
π
or L > √
.
p−1
8
9
10
(3.63)
(3.64)
13
16
17
18
19
q
p−1
aL (x) = ξL
21
22
24
25
λ1 > 0,
31
33
34
1
=
0
(3.65)
(3.66)
14
15
16
17
18
19
21
23
24
25
27
29
Hence for the map L from
to
we know that
inverse of L. This implies that L−1 wL is well defined.
Then we have the following theorem
L2 (I )
(3.67)
L−1
exists, where
L−1
is the
30
31
32
33
34
35
T HEOREM 3.14. Assume that L >
r = 2,
39
L
√π
p−1
36
and either
37
38
wL L
−1
wL dy > 0
(3.68)
0
40
39
40
41
or
42
42
r = p + 1.
(3.69)
44
45
12
28
j = 2, 3, . . . .
λj < 0,
H 2 (I )
38
43
10
26
p−1
+ pwL φ.
37
41
9
22
35
36
8
20
wLr (Lx) dx.
It is proved [89] that L has the spectrum
30
32
wL (Lx),
L[φ] = φ − φ
27
29
qr
1+s− p−1
ξL
Before stating our results, we first introduce some notation. Let I = (0, L) and φ ∈
H 2 (I ). We define the following operator:
26
28
7
13
Thus all non-monotone steady-state solutions are linearly unstable. Therefore we focus
our attention on monotone solutions. There are two monotone solutions—the monotone
increasing one and the monotone decreasing one. Since these two solutions differ by reflection, we consider the monotone decreasing function only. This solution is then called
π
u and it has the least energy among all positive solutions of (3.60), see [60]. If L √p−1
,
then wL = 1. We also denote the corresponding solutions to (3.59) by
20
23
5
11
ax (x, t0 ) = 0 for all (x, t) ∈ (0, 1) × [t0 , +∞).
12
15
4
6
The stability of steady-state solutions to (3.59) has been a subject of study in the last few
years. A recent result of [56] (see Theorem 1.1 of [56]) says that a stable solution to (3.59)
must be asymptotically monotone. More precisely, if (A(x, t), ξ(t)), t 0 is a solution to
(3.59) that is linearly neutrally stable, then there is a t0 > 0 such that
11
14
2
3
6
7
1
43
44
Then (aL , ξL ) (given by (3.65)) is a linearly stable steady state to (3.59) for τ small.
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Existence and stability of spikes
1
2
3
This theorem reduces the issue of stability to the computation of the integral
4
5
6
7
8
539
1
2
L
3
wL L−1 wL dy.
4
0
5
6
This integral is quite difficult to compute for general L.
For τ finite, we have the following theorem.
7
8
9
10
11
12
13
14
15
16
17
18
19
9
π
T HEOREM 3.15. Let (3.68) be true and L > √p−1
. Then there exists a unique τc > 0
such that for τ < τc , (aL , ξL ) is stable and for τ > τc , (aL , ξL ) is unstable. At τ = τc , there
exists a unique Hopf bifurcation. Furthermore, the Hopf bifurcation is transversal, namely,
we have
22
23
24
25
(3.70)
28
29
30
33
34
35
36
37
38
39
40
41
42
43
44
45
13
15
16
18
where λR is the real part of the eigenvalue.
19
20
Using Weierstrass p(z) functions and Jacobi elliptic integrals, one can show that
L
−1
0 wL L wL dy > 0 for all L > π in the cases r = 2, p = 2, 3. The original Gierer–
Meinhardt system ((p, q, r, s) = (2, 1, 2, 0)) falls into this class. Thus for the shadow system of the original Gierer–Meinhardt system, we have a complete picture of the stability
of (aL , ξL ) for any τ > 0 and any L > 0, by the following theorem
21
22
23
24
25
26
π
T HEOREM 3.16. Assume that L > √p−1
and r = 2, p = 2 or 3. Then there exists a unique
τc > 0 such that for τ < τc , (aL , ξL ) is stable and for τ > τc , (AL , ξL ) is unstable. At
τ = τc , there exists a Hopf bifurcation. Furthermore, the Hopf bifurcation is transversal.
31
32
12
17
26
27
11
14
dλR > 0,
dτ τ =τc
20
21
10
27
28
29
30
31
Theorem 3.16 gives a complete picture of the stability of non-trivial monotone solutions
π
we necessarily have wL ≡ 1. Combining this with the
in terms of L since for L √p−1
results of [56], we have completely classified stability and instability of all steady-state
solutions for all > 0 for the shadow system of the original Gierer–Meinhardt system.
We do not know if the Hopf bifurcation in Theorem 3.15 is subcritical or supercritical. This is related to another interesting question: is there time-periodic solution
(a(x, t), ξ(x, t)) to (3.59) at the Hopf bifurcation point τ = τc ? If so, is it stable or unstable?
We can also extend this idea to general domains in RN , N 2. Namely we consider
⎧
p
⎪
a = a − a + aξ q , x ∈ ΩL ,
⎪
⎨ t
τ ξt = −ξ + ξ −s |Ω1L | ΩL a r ,
⎪
⎪
⎩
a > 0, ∂a
∂ν = 0 on ∂ΩL ,
32
33
34
35
36
37
38
39
40
41
42
t > 0,
43
(3.71)
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1
2
3
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J. Wei
where we have scaled the into the domain through ΩL = 1 Ω. In this case, let us assume
that ΩL ⊂ RN is a smooth and bounded domain, and the exponents (p, q, r, s) satisfy the
following condition
4
p > 1,
q > 0,
r > 0,
6
s 0,
qr
γ :=
> 1,
(p − 1)(s + 1)
6
7
and p is subcritical:
9
1<p<
10
11
13
14
15
8
9
N +2
N −2
if N 3;
1 < p < +∞
if N = 2.
10
11
The steady state solution of (3.71) is given by
q
qr
1
1+s− p−1
p−1
a=ξ
u, ξ
=
ur
|ΩL | ΩL
12
13
(3.72)
16
17
19
20
u − u + up = 0,
∂u
∂ν = 0
18
u > 0 in ΩL ,
on ∂ΩL .
(3.73)
E[wL ] =
24
25
23
inf
u∈H 1 (ΩL ), u≡0
E[u]
(3.74)
26
27
2
2
Ω (|∇u| + u )
.
E[u] = L
2
( ΩL up+1 ) p+1
29
30
28
29
30
31
35
36
37
38
39
40
31
The corresponding steady-state solution to the shadow system (3.71) is denoted by
q
qr
1+s− p−1
1
p−1
=
wr .
(3.75)
a L = ξL wL , ξL
|ΩL | ΩL L
43
44
45
32
33
34
35
36
Let
37
L[φ] = φ − φ
p−1
+ pwL φ.
38
39
40
Then we have the following lemma whose proof is similar to Lemma 3.2.
41
42
24
25
28
34
20
22
where
27
33
19
21
We again consider the minimizer solution wL (x) which satisfies (3.73) and
23
32
15
17
21
26
14
16
where u is a solution of the following problem:
18
22
3
5
7
12
2
4
5
8
1
41
L EMMA 3.17. Consider the following eigenvalue problem
Lφ = λφ, in ΩL ,
∂φ
on ∂ΩL .
∂ν = 0
42
43
44
(3.76)
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Existence and stability of spikes
1
541
Then λ1 > 0 and λ2 0.
1
2
2
3
3
We now put two important assumptions:
We first assume that
4
4
5
5
6
(A1) L
7
8
−1
6
exists.
7
8
Under (A1), we assume that
9
10
10
wL L−1 wL > 0.
(A2)
11
9
11
ΩL
12
12
13
13
We can now state the following theorem
14
14
15
16
15
T HEOREM 3.18. Assume that either
16
17
17
r = p + 1,
18
and (A1) holds,
18
19
20
19
or
20
21
22
21
r = 2,
22
and (A1) and (A2) hold.
23
24
25
26
27
28
29
30
31
32
23
Then (aL , ξL ) is linearly stable for τ small.
In the case of r = 2, there exists a unique τ = τc such that (aL , ξL ) is stable for τ < τc ,
unstable for τ > τc , and there is a Hopf bifurcation at τ = τc . Furthermore, the Hopf
bifurcation is transversal.
The proof of Theorem (3.18) is similar to the one-dimensional case.
It remains an interesting and difficult question as to verify (A1) and (A2) analytically. If
L is large, the assumption (A1) is verified in [76] and assumption (A2) holds true if
33
34
35
36
37
38
39
1<p<1+
4
.
N
This recovers the results of [77].
It is difficult to verify (A1) and (A2) in general domains. One may ask: does (A1) hold
true for generic domains?
27
28
29
30
31
32
34
35
36
37
38
39
41
3.6. The stability of boundary spikes for the Robin boundary condition
43
45
26
40
41
44
25
33
(3.77)
40
42
24
42
43
The stability of least energy solution in the Robin boundary condition case is quite complicated. We state the following result which deals with one-dimensional case only:
44
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2
3
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J. Wei
T HEOREM 3.19. (See [45].) Consider the following
1
⎧
p
at = 2 axx − a + aξ q , 0 < x < 1, t > 0,
⎪
⎪
⎪
1
⎨
τ ξt = −ξ + ξ −s 0 a r dx,
⎪
⎪
⎪
⎩ a > 0, ax (0, t) + λa(0, t) = ax (1, t) + λa(1, t) = 0,
hx (0, t) = hx (1, t) = 0.
2
3
4
(3.78)
6
7
8
9
10
11
8
Assume that r = 2, 1 < p 3 or r = p + 1, 1 < p < +∞. Then for each λ ∈ (0, 1) the
least energy solution is stable for τ < τ1 and unstable for τ > τ1 . At τ1 , there is a Hopf
bifurcation.
12
13
14
15
16
17
18
19
22
23
24
The main idea of the proof is similar to that of Theorem 3.14. Here we have to study an
NLEP on a half line with Robin boundary condition:
⎧
∞
⎨ φ − φ + pw p−1 φ − γ (p − 1) 0 wx0 φ w p = αφ,
∞ 2
x0
x0
0 wx0
⎩ φ (0) − λφ(0) = 0
27
28
29
30
33
34
35
11
13
14
15
16
0 < y < +∞,
17
(3.79)
18
19
where wx0 = w(y − x0
we need to show that
0
) with w (−x
p−1
0 ) = λw(−x0 ). Let Lx0 (φ) = φ − φ + pwx0 φ. Then
21
22
23
∞
wx0 L−1
x0 (wx0 ) > 0.
24
(3.80)
25
26
∞
By some lengthy computations, we can show that the function 0 wx0 [L−1
x0 (wx0 )] is an
increasing function in x0 when p < 3, and a constant when p = 3, and an decreasing
function when p > 3.
31
32
10
20
25
26
9
12
20
21
5
27
28
29
30
31
R EMARK 3.6.1. An interesting phenomena is the case of 3 < p < 5. In this case, one
can show that there exists a a0 ∈ (0, 1) such that the boundary spike is stable when a ∈
(0, a0 ) and unstable when a ∈ (a0 , 1). It is quite interesting to see that the Robin boundary
condition can also introduce some instability.
32
33
34
35
36
36
37
37
38
39
38
4. Full Gierer–Meinhardt system: One-dimensional case
40
41
42
43
44
45
39
40
In this section, we study the full Gierer–Meinhardt system in the one-dimensional case.
Unlike the shadow system case, where one can reduce the existence of solutions to a variational elliptic problem, there is no variational structure for the full Gierer–Meinhardt system. This is the major problem, which is also the source of all interesting new phenomena.
We begin with the steady-state problem in the full space case.
41
42
43
44
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Existence and stability of spikes
1
543
4.1. Bound states: the case of Ω = R1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
2
Let Ω = R1 . By a change of variables the steady-state problem for (GM) can be conveniently written as follows
⎧
ap
⎪
in R1 ,
⎨ a − a + hq = 0, a > 0
r
(4.1)
h − σ 2 h + ahs = 0, h > 0 in R1 ,
⎪
⎩
a(x), h(x) → 0
as |x| → +∞
20
21
22
23
24
27
28
29
σ2 =
32
33
34
35
36
37
38
39
D
42
45
8
9
14
The existence of multiple spikes solutions to (4.1) is referred to as “symmetry-breaking”
phenomena. This was proved in [12] (by dynamical system techniques) and [7] (by PDE
methods). We will sketch the PDE methods in Section 5.1.
15
16
17
18
T HEOREM 4.1. (See [7,12].) For each fixed positive integer k, there exists σk > 0 such
that problem (4.1) has a solution (a , h ) with the following properties
19
20
21
k
ck w x − ξjσ
a (x) ∼
σ
22
(4.2)
j =1
23
24
25
where ck > 0 is a generic constant and
ξjσ = j −
26
1
1
k+1
log + O log log
,
2
σ
σ
27
j = 1, . . . , k.
(4.3)
28
29
30
4.2. The bounded domain case: Existence of symmetric K-spikes
Without loss of generality, we may assume that Ω = (−1, 1). We consider the following
elliptic system
⎧
ap
2 ⎪
⎨ a − a + hq = 0, −1 < x < 1,
r
(4.4)
Dh − h + ahs = 0, −1 < x < 1,
⎪
⎩ a (±1) = h (±1) = 0.
31
32
33
34
35
36
37
38
39
40
In this case, the existence of multiple-peaked solutions was first obtained by I. Takagi in
[67].
43
44
7
13
40
41
6
12
1.
30
31
5
11
2
25
26
4
10
where
18
19
3
41
42
43
T HEOREM 4.2. (See [67].) Fix any positive integer K. If √ sufficiently small, there exD
ists a K-peaked solution (a,K , hK ) to (4.4) such that (a,K , h,K ) has exactly K local
44
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
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J. Wei
maximum points −1 < x1 < x2 < · · · < xK < 1 which are equally distributed. In fact, we
have
2j − 1
,
xj = −1 +
K
18
19
20
j = 1, . . . , K.
4
5
Takagi’s proof uses the symmetry of the problems: by reflection, one can reduce the
existence of multiple symmetric spikes solutions to studying the existence of one boundary
spike solution. Namely, we just need to study the following system
⎧ 2 p
1
a − a + ahq = 0, 0 < x < 2K
,
⎪
⎪
⎪
r
⎪
⎨ Dh − h + a s = 0, 0 < x < 1 ,
h
2K
q
x
⎪
p−1
⎪ a(x) ∼ ξ
w( ), h(0) = ξ,
⎪
⎪
⎩ 1
1
a (0) = a ( 2K ) = h (0) = h ( 2K
) = 0.
27
28
29
30
35
36
37
38
39
40
41
42
43
44
45
13
14
15
17
18
19
20
21
22
4.3. The bounded domain case: existence of asymmetric K-spikes
23
24
In the bounded domain case, as D is getting smaller, more and more new solutions appear.
By using the same matched asymptotic analysis in [34], M. Ward and Wei in [70] discovered that for D < DK , where DK is given by (4.67) below, problem (4.4) has asymmetric
K-peaked steady-state solutions. Such asymmetric solutions are generated by two types
of peaks-called type A and type B, respectively. Type A and type B peaks have different
heights. They can be arranged in any given order
25
26
27
28
29
30
31
ABAABBB...ABBBA...B
33
34
9
16
For the one boundary spike solution, one can use the Implicit Function Theorem,
since the linearized operator is invertible in the space of functions with Neumann boundary conditions. (The last statement follows from the fact that the kernel of the operator
− 1 + pw p−1 consists exactly those of partial derivatives of w. See Lemma 3.2.)
31
32
8
12
(4.5)
24
26
7
11
22
25
6
10
21
23
2
3
16
17
1
32
33
to form an K-peaked solution. The existence of such solutions is surprising. It shows that
the solution structure of (4.4) is much more complicated than one would expect. The stability of such asymmetric K-peaked solutions is also studied in [70], through a formal approach. We remark that asymmetric patterns can also be obtained for the Gierer–Meinhardt
system on the real line, see [12].
In this and next section, we present a rigorous and unified theoretic foundation for the
existence and stability of general K-peaked (symmetric or asymmetric) solutions. In particular, the results of [34] and [70] are rigorously established. Moreover, we show that if
the K peaks are separated, then they are generated by peaks of type A and type B, respectively. This implies that there are only two kinds of K-peaked patterns: symmetric
K-peaked solutions constructed in [67] and asymmetric K-peaked patterns constructed in
[70].
34
35
36
37
38
39
40
41
42
43
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Existence and stability of spikes
1
2
3
4
5
6
7
8
9
10
11
12
545
The existence proof is based on Lyapunov–Schmidt reduction. Stability is proved by first
separating the problem into the case of large eigenvalues which tend to a non-zero limit
and the case of small eigenvalues which tend to zero in the limit → 0. Large eigenvalues
are then explored by studying non-local eigenvalue problems using results in Section 3.3
and employing an idea of Dancer [8]. Small eigenvalues are calculated explicitly by an
asymptotic analysis with rigorous error estimates.
In this section, we present the existence part.
Before we state our main results, we introduce some notation. Let GD (x, z) be the Green
function of
DGD (x, z) − GD (x, z) + δz i(x) = 0 in (−1, 1),
(4.6)
GD (−1, z) = GD (1, z) = 0.
13
14
16
GD (x, z) =
17
θ
sinh(2θ)
θ
sinh(2θ)
cosh[θ (1 + x)] cosh[θ (1 − z)],
−1 < x < z,
cosh[θ (1 − x)] cosh[θ (1 + z)],
z<x<1
(4.7)
27
28
29
30
31
32
33
34
37
38
39
40
41
42
43
44
45
7
8
9
10
11
12
16
19
20
θ = D −1/2 .
21
22
We set
23
24
1 − √1 |x−z|
KD |x − z| = √ e D
,
2 D
(4.8)
p−1
(p−1)(s+1)−qr
r
ξ := w (z) dz
.
) ∈ (−1, 1)K ,
GD (t) = GD (ti , tj ) .
29
30
31
32
34
36
let
37
(4.10)
as ∇ti . When i = j , we can define ∇ti G(ti , tj ) in the classical way.
When i = j , KD (|ti − tj |) = KD (0) =
28
35
We introduce several matrices for later use: For t = (t1 , . . . , tK
∂
∂ti
27
33
(4.9)
R
Let us denote
25
26
to be the singular part of GD (x, z) and by GD = KD − HD we define the regular part HD
of GD . Note that HD is C ∞ in both x and z.
0 < 1 be K points in (−1, 1) and w be the unique
Let −1 < t10 < · · · < tj0 < · · · < tK
solution of (2.8).
Put
35
36
6
17
22
26
5
18
21
25
4
15
where
20
24
3
14
18
23
2
13
We can calculate explicitly
15
19
1
1
√
2 D
is a constant and we define
∂ H (x, ti ).
∇ti GD (ti , ti ) := − ∂x x=ti
38
39
40
41
42
43
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45
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546
1
J. Wei
Similarly, we define
1
2
3
∇ti ∇tj GD (ti , tj ) =
4
5
6
7
10
11
14
N
16
17
if i = j .
3
(4.11)
qr −s
GD ti0 , tj0 ξ̂j0 p−1 = ξ̂i0 ,
6
7
8
(4.12)
(4.13)
25
14
15
i = 1, . . . , N.
(4.14)
20
B = (bij ).
21
(4.15)
24
25
(4.16)
29
where σ (B) is the set of eigenvalues of B.
30
30
31
31
R EMARK 4.3.1. Since the matrix B is of the form GD D, where GD is symmetric and D is
a diagonal matrix, it is easy to see that the eigenvalues of B are real.
34
37
40
qr
−s
GD (ti , tj )ξˆj p−1 = ξ̂i ,
35
36
37
39
i = 1, . . . , K.
(4.17)
j =1
41
40
41
42
45
33
38
K
39
44
32
34
By the assumption (H2) and the implicit function theorem, for t = (t1 , . . . , tK ) near
0 ), there exists a unique solution ξ̂ (t) = (ξ̂ (t), . . . , ξ̂ (t)) for the following
t0 = (t10 , . . . , tK
1
K
equation
38
43
27
28
29
36
22
26
p−1
∈
/ σ (B),
qr − s(p − 1)
28
35
17
23
27
33
16
19
Our second assumption is the following:
(H2) It holds that
26
32
11
13
23
24
10
18
qr −s−1
bij = GD ti0 , tj0 ξ̂j0 p−1
,
22
9
12
Next we introduce the following matrix
21
4
5
j =1
18
20
∇ti ∇tj GD (ti , tj )
if i = j ,
We now have our first assumption:
(H1) There exists a solution (ξ̂10 , . . . , ξ̂N0 ) of the following equation
15
19
2
∇GD (t) = ∇ti GD (ti , tj ) ,
∇ 2 GD (t) = ∇ti ∇tj GD (ti , tj ) .
9
13
∂ ∂ ∂x x=ti ∂y y=ti HD (x, y)
Now the derivatives of G are defined as follows:
8
12
F:chipot506.tex; VTEX/Rita p. 60
42
Set
43
H(t) = ξ̂i (t)δij .
44
(4.18)
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Existence and stability of spikes
1
547
We define the following vector field:
2
1
2
F (t) = F1 (t), . . . , FK (t) ,
3
3
4
5
4
where
5
6
6
7
Fi (t) =
8
K
∇ti GD (ti , tl )ξ̂l
qr
p−1 −s
7
8
l=1
9
9
qr
p−1 −s
10
= −∇ti HD (ti , ti )ξ̂i
11
+
∇ti GD (ti , tl )ξ̂l
qr
p−1 −s
10
,
11
l=i
12
12
i = 1, . . . , K.
13
(4.19)
14
15
16
17
14
Set
15
16
M(t) = ξ̂i−1 ∇tj Fi (t) .
(4.20)
18
19
20
21
19
20
21
F (t0 ) = 0,
det M(t0 ) = 0.
23
24
(4.21)
30
∇tj ξ̂i =
33
34
36
38
39
42
∇tj ξ̂i =
45
32
33
34
35
36
qr
−s
p−1
N
37
GD (ti , tl )ξ̂l
qr
p−1 −s−1
qr
p−1 −s
∇tj ξ̂l + ∇tj GD (ti , tj )ξ̂j
38
.
l=1
∇tj ξ̂i =
39
40
41
For i = j , we have
43
44
31
For i = j , we have
40
41
30
l=1
37
27
29
K
qr −s
∂ GD (ti , tl ) ξ̂lp−1 .
+
∂tj
32
26
28
K
qr
−s−1
qr
−s
GD (ti , tl )ξ̂lp−1
∇tj ξ̂l
p−1
l=1
31
35
24
25
Let us now calculate M(t0 ): Therefore we first compute the derivatives of ξ̂ . It is easy to
see that ξ̂ (t) is C 1 in t. We can calculate:
28
29
22
23
(4.22)
25
27
17
18
Our final assumption concerns the vector field F (t).
0 ):
(H3) We assume that at t0 = (t10 , . . . , tK
22
26
13
42
K
K
qr
qr −s
∂ qr
p−1 −s−1
−s
GD (ti , tl )ξ̂l
∇ti ξ̂l +
GD (ti , tl ) ξ̂lp−1
p−1
∂ti
l=1
l=1
43
44
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F:chipot506.tex; VTEX/Rita p. 62
J. Wei
1
=
2
K
qr
qr
−s−1
−s
qr
−s
GD (ti , tl )ξ̂lp−1
∇ti ξ̂l + ∇ti GD (ti , ti )ξ̂ip−1
p−1
1
2
l=1
3
4
+
5
6
K
∇ti GD (ti , tl )ξ̂l
qr
p−1 −s
3
4
,
5
6
l=1
7
8
9
10
11
7
since ∂t∂ i GD (ti , ti ) = 2∇ti GD (ti , ti ).
Note that
∇tj GD (ti , tj ) = (∇GD )T .
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
12
Therefore if we denote the matrix
∇ξ = (∇tj ξ̂i )
(4.23)
25
26
27
28
29
30
31
32
−1
qr
qr
qr
−s−1
−s
− s GD H p−1
∇ξ(t) = I −
(∇GD )T H p−1
p−1
K
Fj (t) .
+O
18
19
20
21
(4.24)
j =1
24
We can compute M(t0 ) by using (4.24):
qr
−s
M t0 = H−1 ∇ 2 GD H p−1
qr
qr
−s−1
− s ∇GD H p−1
+ H−1
p−1
−1
qr
qr
qr
−s−1
−s
p−1
− s GD H
× I−
(∇GD )T H p−1 .
p−1
37
38
25
26
27
28
29
30
(4.25)
41
42
43
44
45
31
32
33
The existence result is as follows
34
35
T HEOREM 4.3. (See [84].) Assume that assumptions (H1), (H2) and (H3) are satisfied.
,
Then for 1, problem (4.4) has an K-peaked solution which concentrates at t1 , . . . , tK
or more precisely:
39
40
22
23
35
36
15
17
33
34
14
16
then we have
23
24
13
a (x) ∼
K
j =1
q
p−1
ξ
h ti ∼ ξ ξ̂i0 ,
ti → ti0 ,
x − tj
0 q
,
ξ̂j p−1 w
36
37
38
39
(4.26)
40
41
42
i = 1, . . . , K,
i = 1, . . . , K.
(4.27)
43
44
(4.28)
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Existence and stability of spikes
1
2
3
4
549
R EMARK 4.3.2. In the case of symmetric K-peaked solutions, conditions (H2) and (H3)
are not needed, as in the construction of solutions one can restrict the function space to the
class of symmetric functions (see for example [67]). Note that for small (and not only in
the limit → 0) the peaks are placed equidistantly.
5
6
7
10
11
12
13
14
15
16
17
18
19
22
23
24
25
28
31
32
33
36
37
38
39
18
(4.30)
12
13
14
15
16
17
19
22
23
where c(D) is some positive constant depending on D only and the function b(z) is given
by
24
25
26
tanhα (z)
,
b(z) :=
cosh(z)
(p − 1)
α :=
.
qr − (s + 1)(p − 1)
(4.31)
27
28
29
The idea now is that we fix l and try to find another l¯ = l such that the following holds
l
b √
D
¯ l
=b √
,
D
30
31
0 < l < l¯ < 1,
32
(4.32)
33
34
¯ This shows that if there exists a solution to (4.32), we
which will imply that h(l) = h(l).
¯ In other words, we may match up solutions of (4.29) in
may match up h(l) and h(l).
different intervals.
It turns out that for D small, (4.32) is always solvable. Now (4.32) has to be solved along
with the following interval constraint:
35
36
37
38
39
40
K1 l + K2 l¯ = 1,
K1 + K2 = K.
(4.33)
41
42
For a solution l of (4.60) and (4.33) and j = 1, . . . , K we define
43
44
45
11
21
l
,
h(l) ∼ c(D)b √
D
42
43
10
20
40
41
7
Then the single interior symmetric spike solution is considered which was constructed by
I. Takagi [67]. By some simple computations based on (4.6), we have that
34
35
6
9
29
30
4
8
26
27
3
In [70], by using matched asymptotic analysis, Ward and the first author constructed
such solutions and studied their stability. We now summarize their main ideas. First (4.4)
is solved in a small interval (−l, l):
⎧
p
2 a − a + ahq = 0
in (−l, l),
⎪
⎪
⎪
⎨
ar
Dh − h + hs = 0
in (−l, l),
(4.29)
⎪
a(x)
>
0,
h(x)
>
0
in (−l, l),
⎪
⎪
⎩ a (−l) = a (l) = h (−l) = h (l) = 0.
20
21
2
5
R EMARK 4.3.3. Our results here can be applied to give a rigorous proof for the existence
and stability of K-peaked solutions consisting of peaks with different heights.
8
9
1
44
lj = l
or
lj = l¯
(4.34)
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1
2
3
4
0
t − t 0 = lj + lj −1 ,
j
j +1
6
7
9
10
11
12
13
14
15
J. Wei
where the number of j ’s such that lj = l is K1 (and consequently the number of j ’s such
that lj = l¯ is K2 ). We call the small spike with lj = l type A and the large spike with lj = l¯
type B.
Then we choose tj0 such that
5
8
F:chipot506.tex; VTEX/Rita p. 64
24
25
26
27
28
29
30
31
32
33
36
It is difficult to check (H1) directly. Instead we note that
(See [34] and [70].) More precisely, we calculate
⎛
⎞
.. ..
.
.
0 ⎟
⎜ γ1 β1 0
⎜
⎟
.
.
⎜β γ β
⎟
.. ..
⎜ 1 2 2
⎟
⎜
⎟
⎜ .. .. .. .. .. .. ⎟
√
⎜
⎟
.
.
.
.
.
.
−1
⎟
GD
= (aij ) = 2 D ⎜
⎜. .
⎟
⎜ .. .. β
0 ⎟
j −1 γj βj
⎜
⎟
⎜
⎟
⎜ .. .. .. .. .. .. ⎟
.
.
.
.⎟
⎜ . .
⎝
⎠
.
.
0 . . . . 0 βN −1 γN
−1
GD
is a tridiagonal matrix.
43
44
45
13
14
15
19
21
23
24
25
26
27
28
29
30
31
32
33
34
35
γj = coth(θj −1 + θj ) + coth(θj + θj +1 ),
42
12
22
where
38
41
11
20
γ1 = coth(θ1 + θ2 ) + tanh(θ1 ),
40
10
18
(4.36)
37
39
9
17
lj
θj = √ .
D
34
35
8
16
20
23
4
7
where
19
22
3
6
0
where t00 = −1, tK+1
= 1.
By using matched asymptotics, we now have K1 type A and K2 type B peaks. This ends
the short review of the ideas in [70]. Let us now use Theorem 4.3 to give a rigorous proof
of results of [70]. In order to apply Theorem 4.3, we have to check the three assumptions
(H1), (H2) and (H3).
To this end, let us set
√
ξ̂j0 = (2 D) tanh(θj ), j = 1, . . . , K,
(4.35)
18
21
2
5
j = 0, . . . , K,
16
17
1
36
37
j = 2, . . . , K − 1,
38
39
γK = coth(θK−1 + θK ) + tanh(θK ),
βj = − csch(θj + θj +1 ),
40
j = 1, . . . , N − 1
and θj was defined in (4.36). Note that
√
aij = 2 D(βj δi(j −1) + γj δij + βj +1 δi(j +1) ).
41
42
43
44
(4.37)
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Existence and stability of spikes
1
551
Verifying (4.14) amounts to checking the following identity
1
2
2
N
3
4
aij ξ̂j0 = ξ̂i0
qr
p−1 −s
3
(4.38)
,
j =1
5
5
6
7
8
9
10
11
12
13
6
which is an easy exercise.
It remains to verify (H2) and (H3).
To this end, we need to know the eigenvalues of B and M. In the same way as for the
matrix GD , one can show that B −1 is a tridiagonal matrix. However, it is almost impossible
to obtain an explicit formula for the eigenvalues. Numerical software for solving eigenvalue problems of large matrices is indispensable. Then (H2) has to be checked explicitly.
Numerical computations in [70] do suggest that assumption (H3) is always satisfied.
14
15
16
17
18
19
20
21
22
23
24
T HEOREM 4.4. (See [84].) Suppose that for sufficiently small, there are solutions
(a , h ) of (4.4) such that
a (x) ∼
K
q
p−1
ξ
j =1
(4.39)
h ti ∼ ξ ξ̂i ,
17
18
19
20
21
22
23
24
26
30
i = 1, . . . , K,
(4.40)
31
32
where ξ is given by (4.9),
33
34
34
ξ̂i → ξ̂i0 > 0,
35
ti → ti0 ,
ti0 = tj0 ,
i = j, i, j = 1, . . . , K.
(4.41)
36
35
36
Then necessarily, we have
38
0
¯
li := ti0 − ti−1
∈ {l, l},
39
37
38
i = 1, . . . , K,
(4.42)
where = −1, l and l¯ satisfy (4.32) and (4.33) with K1 being the number of i’s for which
li = l and K2 being the number of i’s for which li = l¯ (hence K1 + K2 = K).
t00
43
45
13
29
32
44
12
28
31
42
11
27
and
30
41
10
25
x − tj
q
,
ξ̂j p−1 w
28
40
9
16
A natural question is the following: Are all K-peaked solution generated by two types of
peaks as the solutions which were constructed in [70]?
Our next theorem gives an affirmative answer. It completely classifies all K-peaked
solutions, provided that the K peaks are separated.
27
37
8
15
4.4. Classification of asymmetric patterns
26
33
7
14
25
29
4
39
40
41
42
43
Theorem 4.4 shows that an K-peaked solution must be generated by exactly two types
¯ This shows that the
of peaks—type A with shorter length l and type B with larger length l.
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J. Wei
solutions constructed in [70] (through a formal approach) exhaust all possible separated
K-peaked solutions. In particular, it shows that there are at most 2K K-peaked solutions.
If the assumptions (H1)–(H3) are met, then there are exactly 2K K-peaked solutions.
4
5
8
9
10
11
12
13
14
15
16
17
18
19
20
23
24
25
5
a = ξ
â ,
7
h = ξ ĥ
8
9
where ξ is defined at (4.9). Hence (â , ĥ ) satisfies
⎧
⎨ 2 â − â +
p
â
q
ĥ
= 0,
10
11
−1 < x < 1,
12
(4.43)
⎩ Dĥ − ĥ + c âr = 0, −1 < x < 1,
s
ĥ
where c is defined as c = ( R w r )−1 .
Now (4.39) and (4.40) imply that
15
16
17
18
K
x − tj
q
,
â ∼
ξ̂j p−1 w
ĥ tj = ξ̂j .
19
(4.44)
j =1
28
tj → tj0 ,
22
23
24
j = 1, . . . , K.
25
26
We see that ĥ → h0 (x) where h0 (x) satisfies
K
29
Dh0 − h0 +
30
h0 (−1) = h0 (1) = 0.
31
32
27
28
qr
0 p−1
j =1 (ξ̂j )
−s
δ(x − tj0 ) = 0,
−1 < x < 1,
29
(4.45)
32
In other words, we have
33
34
h0 (x) =
36
34
K
0 qr −s
ξ̂j p−1 GD x, tj0 .
(4.46)
39
37
Since h0 (tj0 ) = ξ̂j0 , j = 1, . . . , K, we have from (4.46) that (ξ̂10 , . . . , ξ̂K0 ) must satisfy the
following identity:
40
41
42
43
38
39
40
K
qr −s
GD ti0 , tj0 ξ̂j0 p−1 = ξ̂i0 ,
j =1
44
45
35
36
j =1
37
38
30
31
33
35
20
21
Letting → 0, we assume that
ξ̂j → ξ̂j0 ,
13
14
26
27
3
6
q
p−1
21
22
2
4
P ROOF OF T HEOREM 4.4. First we make the following scaling
6
7
1
41
i = 1, . . . , K.
(4.47)
42
43
44
This is the same as (4.14).
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Existence and stability of spikes
1
Define
1
2
3
ã,j = â χ
4
x − tj0
7
8
2
3
4
r̃0
5
6
553
5
where r̃0 is a very small number. Then ã,j is supported in the interval Ij = (−r̃0 + tj , r̃0 +
tj ). We may choose r̃0 so small that Ii ∩ Ij = ∅ for i = j . Then
9
â =
11
K
11
12
13
13
Now we multiply the first equation in (4.43) by ã,j
and integrate over (−1, 1). We
obtain
16
0=
18
19
20
=
21
22
23
= −2
24
p â â
p
q
Ij
25
q(p + 2)
=
p+1
26
27
ĥ
28
Ij
−
18
19
20
21
p+1 ĥ
+ e.s.t.
q+1
ĥ
22
q â
23
24
25
p+1
â
ĥ
q+1 ĥ
+ e.s.t.
(4.48)
ĥ (x) = c
33
30
31
1
−1
GD (x, z)
âr
32
ĥs
33
34
34
and thus for x
36
∈ Ij ,
37
38
ĥ (x) =
39
44
45
35
36
K
37
qr −s
GD x, tk ξ̂k p−1 + O()
38
39
k=1
40
43
27
29
32
42
26
28
By the equation for ĥ , we have that
31
41
15
17
29
35
14
16
p 1 aˆ p â
ã,j
q ã,j −
q
ĥ
−1
hˆ
p â
â + e.s.t.
−2
q
Ij ĥ
17
30
8
10
ã,j + e.s.t.
j =1
12
15
7
9
10
14
6
40
41
and
42
K
qr −s
∇tj GD tj , tk ξ̂k p−1 + O().
Ĥ tj =
k=1
43
(4.49)
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J. Wei
Substituting (4.49) into (4.48) and using (4.44), we obtain the following identity
1
2
3
2
K
4
7
8
9
10
qr −s
tj , tk ξ̂k p−1 = o(1)
3
(4.50)
k=1
5
6
∇tj GD
5
6
and hence
7
K
qr −s
∇t 0 GD tj0 , tk0 ξ̂k0 p−1 = 0,
j
8
j = 1, . . . , K,
(4.51)
13
14
15
16
17
18
19
20
21
22
11
which is the same as (4.21).
Note that by the expression for h0 in (4.46), (4.51) is equivalent to the following
h0 tj0 + + h0 tj0 − = 0, j = 1, . . . , K,
(4.52)
where h0 (tj0 +) is the right-hand derivative of h0 at tj0 and h0 (tj0 −) is the left-hand derivative of h0 at tj0 . On the other hand, from the equation for h0 , we have that
qr −s
D h0 tj0 + − h0 tj0 − = − ξ̂j0 p−1 ,
25
26
27
28
29
30
31
32
33
34
35
36
37
j = 1, . . . , K.
(4.53)
40
41
42
45
14
15
16
17
18
20
21
23
qr
1 0 p−1
−s
h0 tj0 + = −h0 tj0 − = −
ξ̂j
< 0,
2D
j = 1, . . . , K.
(4.54)
Since h0 satisfies Dh0 = h0 > 0 in each interval (tj0−1 , tj0 ), j = 2, . . . , K, we see that
there exists a unique point sj −1 ∈ (tj0−1 , tj0 ) such that h0 (sj −1 ) = 0. Since h0 (−1) = 0, by
using symmetry, we see that
24
25
26
27
28
29
30
sj −1 + sj
= tj0 ,
2
j = 1, . . . , K,
(4.55)
Dh0 − h0 + ξ̂j0
qr
p−1 −s
δ t
− tj0
31
32
33
where we take s0 = −1, sK = 1. Let 2lj = sj − sj −1 , j = 1, . . . , K.
Note that on each interval (−lj + tj0 , lj + tj0 ), h0 satisfies
34
35
36
=0
37
38
with Neumann boundary conditions at both ends. Thus from (4.6) it is easy to see that
√
0 qr −s−1
lj
ξ̂j p−1
, j = 1, . . . , K,
(4.56)
= 2 D tanh √
D
43
44
13
22
Solving (4.52) and (4.53), we have that
38
39
12
19
23
24
9
10
k=1
11
12
4
h0 (lj ) =
39
40
41
42
43
ξ̂j0
l
cosh( √j )
D
.
(4.57)
44
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Existence and stability of spikes
1
555
Since h0 is continuous on (−1, 1), we have
1
2
3
2
h0 (l1 ) = h0 (l2 ) = · · · = h0 (lK ).
(4.58)
4
5
6
7
8
9
10
11
12
13
4
Using (4.56) and (4.57), we see that (4.58) is equivalent to
l1
l2
lK
b √
=b √
= ··· = b √
,
D
D
D
5
6
(4.59)
16
17
18
19
20
where the function b was defined in (4.31). Suppose without loss of generality that l1 l2 ,
¯ and that lj ∈ {l, l}
¯ for j = 2, . . . , K.
then we take l1 = l and (4.59) implies that l2 ∈ {l, l}
Thus l must satisfy (4.60) and (4.33).
This finishes the proof of Theorem 4.4.
23
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
10
11
12
13
15
4.5. The stability of symmetric and asymmetric K-spikes
16
In this section, we present the stability of the K-peaked solutions constructed in Theorem
4.3.
To this end, we need to study the following linearized eigenvalue problem
L
24
25
9
14
21
22
7
8
14
15
3
φ
ψ
2 φ − φ + p
=
1
τ Dψ − ψ + r
p−1
a
q
H
ar−1
hs
φ − q
ψ , φ
=
λ
,
ψ
ψ
p
a
q+1
h
ar
hs+1
φ − s
(4.60)
and furthermore
qr
− s min σ > 1
p−1
σ ∈σ (B)
(4.61)
20
22
23
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
(4.62)
41
42
and
σ (M) ⊆ σ Re(σ ) > 0 ,
19
24
T HEOREM 4.5. Let (a , h ) be the solutions constructed in Theorem 4.3. Assume that
1.
(1) (Stability) If
r = p + 1, < p < +∞
18
21
where (a , h ) is the solution constructed in Theorem 4.3 and λ ∈ C—the set of complex
numbers.
We say that (a , h ) is linearly stable if the spectrum σ (L ) of L lies in the left half
plane {λ ∈ C: Re(λ) < 0}. (a , h ) is called linearly unstable if there exists an eigenvalue
λ of L with Re(λ ) > 0. (From now on, we use the notations linearly stable and linearly
unstable as defined above.)
r = 2, < p < 5 or
17
43
44
(4.63)
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J. Wei
there exists τ0 > 0 such that (a , h ) is linearly stable for 0 τ < τ0 .
(2) (Instability) If
5
6
7
8
9
10
11
F:chipot506.tex; VTEX/Rita p. 70
qr
−s
p−1
1
2
3
4
min σ < 1,
σ ∈σ (B)
(4.64)
6
7
there exists τ0 > 0 such that (a , h ) is linearly unstable for 0 τ < τ0 .
(3) (Instability) If there exists
σ ∈ σ (M),
8
9
10
Re(σ ) < 0,
(4.65)
12
13
16
13
14
R EMARK 4.5.1. In the original Gierer–Meinhardt model, (p, q, r, s) = (2, 1, 2, 0) or
(p, q, r, s) = (2, 4, 2, 0). This means that condition (4.61) is satisfied.
17
18
19
20
21
22
23
24
25
28
29
R EMARK 4.5.2. By Theorems 4.3 and 4.5, the existence and stability of K-peaked solutions are completely determined by the two matrices B and M. They are related to the
asymptotic behavior of large eigenvalues which tend to a non-zero limit and small eigenvalues which tend to zero as → 0, respectively. The computations of these two matrices
are by no means easy. We refer to [34] and [70] for exact computations and numerics. Combining the results here and the computations in [34], the stability of symmetric K-peaked
solutions is completely characterized and the following result is established rigorously.
r = p + 1, 1 < p < +∞
(4.66)
36
39
40
D < DK :=
√
K 2 (log(
1
,
√
α + α + 1))2
45
22
23
24
25
27
28
29
31
35
(4.67)
36
37
where α is given by (4.31), then the symmetric K-peaked solution is linearly stable.
(b) (Instability) If
38
39
40
41
D > DK ,
(4.68)
43
44
21
33
41
42
20
34
37
38
19
32
and
34
35
18
30
r = 2, 1 < p < 5 or
32
33
16
26
T HEOREM 4.6. (See [34,84].) Let (a,K , h,K ) be the symmetric K-peaked solutions constructed in [67]. Assume that 1.
(a) (Stability) Assume that 0 < τ < τ0 for some τ0 small and that
30
31
15
17
26
27
11
12
then (a , h ) is linearly unstable for all τ > 0.
14
15
5
42
43
where DK is given by (4.67), then the symmetric K-peaked solution is linearly unstable for all τ > 0.
44
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Existence and stability of spikes
1
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3
557
The proof of Theorem 4.5 consists of two parts: we have to compute both small and
large eigenvalues. For large eigenvalues, we will arrive at the following system of nonlocal eigenvalue problems (NLEPs)
4
5
−1
− qr(I + sB)
7
11
12
13
14
B
R
w
r−1
Φ
R
w
r
−1
6
w = λΦ
p
(4.69)
⎛
φ1
⎜ φ2
⎜
Φ = ⎜.
⎝ ..
9
10
⎞
11
12
⎟ K
⎟
⎟ ∈ H 2 (R) .
⎠
13
14
φK
15
16
18
19
20
21
22
16
By diagonalization, we may reduce it to K NLEPs of the form (3.17). Using the results
of Theorem 3.7, we obtain the stability (or instability) of large eigenvalues.
For the study of small eigenvalues, we need to expand the eigenfunction up to the order
O( 2 ) term. This computation is quite involved. In the end, the matrix B and M will
appear.
A similar stability analysis for the Schnakenberg model has been carried out in [35].
23
28
29
30
37
38
39
40
41
42
43
44
45
21
22
27
28
29
30
31
32
5.1. Bound states: spikes on polygons
33
34
36
20
26
Let us now consider the Gierer–Meinhardt system in a two-dimensional domain. The results are more complicated. To reduce the complexity and grasp the essential difficulties,
we assume that (p, q, r, s) = (2, 1, 2, 0) in this section.
We start with the bound states.
32
35
19
25
31
33
18
24
5. The full Gierer–Meinhardt system: Two-dimensional case
26
27
17
23
24
25
7
8
where B is given by (4.15) and
15
17
3
5
8
10
2
4
Φ − Φ + pw p−1 Φ
6
9
1
34
We first consider the case when Ω = R2 :
⎧
a2
⎪
⎨ a − a + h = 0, a > 0
⎪
⎩
h − σ 2 h + a 2
a(x), h(x) → 0
= 0,
35
36
in R2 ,
R2 ,
h > 0 in
as |x| → +∞.
37
(5.1)
As we will see, a notable feature of this ground-state problem in the plane is the presence of solutions with any prescribed number of bumps in the activator as the parameter σ
gets smaller. These bumps are separated from each other at a distance O(| log log σ |) and
approach a single universal profile given by the unique radial solution of (2.8). The multibump solutions correspond respectively to bumps arranged at the vertices of a k-regular
38
39
40
41
42
43
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2
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5
J. Wei
polygon and at those of two concentric regular polygons. These arrangements with one extra bump at the origin are also considered. This unveils a new side of the rich and complex
structure of the solution set of the Gierer–Meinhardt system in the plane and gives rise to
a number of questions.
Let us set
6
7
8
τσ =
1
k
log
2π
σ
9
10
11
12
13
14
15
16
F:chipot506.tex; VTEX/Rita p. 72
R2
w 2 (y) dy
19
20
21
22
25
26
27
(5.2)
k
lim τσ aσ (x) −
w(x − ξi ) = 0,
σ →0
32
33
34
35
36
37
38
39
42
43
10
11
12
15
16
17
uniformly in x ∈ R2 . Here the points ξi correspond to the vertices of a regular polygon
centered at the origin, with sides of equal length lσ satisfying
1
1
.
lσ = log log + O log log log
σ
σ
18
19
20
21
(5.4)
22
23
Finally, for each 1 j k we have
24
25
lim τσ hσ (ξj + y) − 1 = 0,
26
σ →0
27
28
uniformly on compact sets in y.
29
30
Our second result gives existence of a solution with bumps at vertices of two concentric
polygons.
T HEOREM 5.2. (See [17].) Let k 1 be a fixed positive integer. There exists σk > 0 such
that, for each 0 < σ < σk , problem (5.1) admits a solution (a, h) with the following property:
k
w(x − ξi ) + w(x − ξi∗ ) = 0,
lim τσ aσ (x) −
σ →0
31
32
33
34
35
36
37
38
(5.5)
i=1
39
40
uniformly in x ∈ R2 . Here the points ξi and ξi∗ are the vertices of two concentric regular
polygons. They satisfy
44
45
8
14
(5.3)
i=1
40
41
5
13
30
31
4
9
T HEOREM 5.1. (See [17].) Let k 1 be a fixed positive integer. There exists σk > 0 such
that, for each 0 < σ < σk , problem (5.1) admits a solution (a, h) with the following property:
28
29
3
7
.
23
24
2
6
−1
17
18
1
41
42
43
44
ξj = ρ σ e
2j π
k i
,
ξj∗
2πj
= ρσ∗ e k i ,
j = 1, . . . , k,
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Existence and stability of spikes
1
559
where
1
2
2
1
1
ρσ =
,
log log + O log log log
2πi
σ
σ
|1 − e k |
1
3
4
3
4
5
6
5
and
∗
ρσ = 1 +
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
30
31
8
9
12
11
The method employed in the proof of the above results consists of a Lyapunov–Schmidt
type reduction. The basic idea of solving the second equation in (5.1) for h first and then
working with a non-local elliptic PDE rather than directly with the system. Let T (a 2 ) be
the unique solution of the equation
h − σ 2 h + a 2
=0
h(x) → 0
a2
R2 ,
(5.6)
a − a +
T (a 2 )
34
(5.7)
39
40
41
42
43
44
45
17
18
19
20
21
23
25
26
27
29
31
32
2
1
1
log log |ξj − ξi | 2 log log ,
3
σ
σ
33
for all i = j.
34
35
We look for solutions to (5.7) of the form
37
38
16
30
Fixing m points which satisfy the constraints
35
36
15
28
= 0.
32
33
14
24
for
Equation (5.3) can be solved via sub-super-solution method. Solving the
second equation for h in (5.1) we get h = T (a 2 ), which leads to the non-local PDE for a
a2
13
22
in
as |x| → +∞,
∈ L2 (R2 ).
28
29
7
T HEOREM 5.3. (See [17].) Let k 1 be given. Then there exists solutions which are exactly as those in Theorems 5.1
and 5.2 but with an additional bump at
the origin. More
precisely, with w(x) added to ki=1 w(x − ξi ) in (5.3) and added to ki=1 [w(x − ξi ) +
w(x − ξi∗ )] in (5.5).
24
27
1
1
1
log log + O log log log
.
2πi
σ
σ
|1 − e k |
10
23
26
6
A similar assertion to (5.4) holds for hσ , around each of the ξi and the ξi∗ ’s.
22
25
1
a(x) = (W + φ),
τσ
where W =
36
K
37
w(x − ξj ).
(5.8)
j =1
By using finite-dimensional reduction method, we first solve an auxiliary problem
⎧
+φ)2
∂W
⎨ (W + φ) − (W + φ) + (W
= i,α ciα ∂ξ
1
2
i,α
T ( τσ (W +φ) )
(5.9)
⎩ φ ∂W = 0, i = 1, . . . , m, α = 1, 2.
R2 ∂ξi,α
38
39
40
41
42
43
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J. Wei
Solutions satisfying the required conditions in Theorems 5.1–5.3 will be precisely those
satisfying a non-linear system of equations of the form
3
4
5
6
7
8
9
10
13
ciα (ξ1 , ξ2 , . . . , ξm ) = 0,
i = 1, . . . , m, α = 1, 2,
4
5
6
where for such a class of points the functions ciα satisfy
7
∂ ciα (ξ1 , . . . , ξk ) =
F |ξj − ξi | + iα ,
∂ξiα
8
(5.10)
11
function F : R+ → R is of the form
F (r) =
16
c7 log r
log
1
σ
12
13
14
+ c8 w(r),
15
16
17
18
17
c7 and c8 are universal constants and
19
20
21
iα = O
1
(log σ1 )1+γ
18
19
20
,
21
22
23
24
25
26
27
28
29
30
22
for some γ > 0. Although (5.10) does not have a variationalstructure, solutions of the
problem ciα = 0 are close to critical points of the functional i=j F (|ξj − ξi |). In spite
of the simple form of this functional, its critical points are highly degenerate because of
the invariance under rotations and translations of the problem. Thus, to get solutions using
degree theoretical arguments, we need to restrict ourselves to classes of points enjoying
symmetry constraints. This is how Theorems 5.1–5.3 are established. On the other hand,
we believe strongly that finer analysis may yield existence of more complex patterns, such
as honey-comb patterns, or lattice patterns.
31
32
33
34
35
36
37
R EMARK 5.1.1. Similar method can also be used to prove Theorem 4.1. In that case, we
have
∂ F1 |ξj − ξi | + O σ 1+γ ,
ci (ξ1 , . . . , ξk ) =
∂ξi
i=j
44
45
25
26
27
28
29
30
32
33
34
36
37
38
function F1 : R+ → R is of the form
39
40
F1 (r) = c9 σ r + c10 w(r),
42
43
24
35
(5.11)
40
41
23
31
38
39
9
10
i=j
14
15
2
3
11
12
1
c9 and c10 are universal constants. It is easy to see that the critical points of
ξj |) is non-degenerate (in the class of points with K
j =1 ξj = 0).
41
42
i=j
F1 (|ξi −
43
44
45
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Existence and stability of spikes
1
561
5.2. Existence of symmetric K-spots
1
2
3
4
2
We look for solutions to the stationary GM on a two-dimensional domain with the following form
5
6
7
a (x) ∼
j =1
8
9
10
11
12
13
14
15
16
17
18
19
20
K
x − Pj
ξ,j w
ξ,i
= 1,
→0 ξ,j
6
(5.12)
23
24
25
26
9
10
11
12
for i = j,
(5.13)
29
30
31
32
33
34
35
36
37
38
39
40
41
42
such solutions are called symmetric K-spots. Otherwise, they are called asymmetric Kspots.
In this section, we discuss the existence of symmetric K-spots. It turns out in twodimensional case, we have to discuss two cases: the strong coupling case, D ∼ O(1), and
the weak coupling case, D 1.
We first have the following existence result in the strong coupling case
45
15
16
17
18
19
20
21
T HEOREM 5.4. (See [86].) Let Ω ⊂ R 2 be a bounded smooth domain and D be a
fixed positive constant. Let GD (x, y) be the Green function of −D + 1 in Ω (with
Neumann boundary condition). Let HD (x, y) be the regular part of GD (x, y) and set
hD (P ) = HD (P , P ).
Set
22
23
24
25
26
27
FD (P1 , . . . , PK ) =
K
i=1
HD (Pi , Pi ) −
28
GD (Pj , Pl ).
j =l
Assume that (P1 , . . . , PK ) ∈ Ω K is a non-degenerate critical point of FD (P1 , . . . , PK ).
Then for sufficiently small, problem (GM) has a steady state solution (a , h ) with the
following properties:
x−Pj
0
(1) a (x) = ξ ( K
j =1 w( ) + o(1)) uniformly for x ∈ Ω̄, Pj → Pj , j = 1, . . . , K,
as → 0, and w is the unique solution of the problem (2.8).
(2) h (x) = ξ (1 + O( | log1 | )) uniformly for x ∈ Ω̄, where
1
+ o(1)) 2 log 1 R2 w 2 .
(3) ξ−1 = ( 2π
R EMARK 5.2.1. Theorem 5.4 shows that interior peaks solutions are related to the Green
function (contrast to shadow system case). Thus in the strong coupling case, the peaks are
produced by a different mechanism. It seems that the equation for h controls everything.
43
44
13
14
27
28
7
8
21
22
4
5
where Pj are the locations of the K-spikes and ξ,j is the height of the spike at Pj .
If all the heights are asymptotically equal, i.e.
lim
3
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
R EMARK 5.2.2. In a general domain, the function FD (P) always has a global maximum
point P0 in Ω × · · · × Ω. (A proof of this fact can be found in the Appendix of [86].)
44
45
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562
1
2
3
4
5
6
1
Next, we discuss the weak coupling case. We assume that lim→0 D = +∞. We first
introduce a Green function G0 which we need to formulate our main results.
Let G0 (x, ξ ) be the Green function given by
3
log ⎧
1
⎪
0 (x, ξ ) − |Ω| + δξ (x) = 0 in Ω,
⎨ G
Ω G0 (x, ξ ) dx = 0,
⎪
⎩ ∂G0 (x,ξ )
=0
on ∂Ω
∂ν
8
9
10
11
13
J. Wei
The proof of Theorem 5.4 depends on fine estimates in the finite-dimensional reduction:
the major problem is to sum up the errors of powers in terms of 1 1 .
7
12
19
11
14
1
1
log
− G0 (x, ξ )
2π
|x − ξ |
(5.15)
17
18
19
20
21
22
with
23
24
24
Pi = (Pi,1 , Pi,2 )
25
for i = 1, 2, . . . , K.
25
26
27
26
For P ∈ Ω K we define
27
28
28
29
F0 (P) =
30
31
K
H0 (Pk , Pk ) −
29
G0 (Pi , Pj )
(5.16)
i,j =1,...,K, i=j
k=1
34
35
36
37
38
39
40
32
and
33
34
M0 (P) = ∇P2 F0 (P) .
Here M0 (P) is a (2K) × (2K) matrix with components
1, 2 (recall that Pi,j is the j th component of Pi ).
Set
(5.17)
43
∂ 2 F (P)
∂Pi,j ∂Pk,l , i, k
= 1, . . . , K, j, l =
37
38
39
40
41
1
D = 2,
β
β 2 |Ω|
1
η :=
log .
2π
44
45
35
36
41
42
30
31
32
33
15
16
22
23
9
10
P = (P1 , P2 , . . . , PK )
21
6
8
(5.14)
be the regular part of G0 (x, ξ ).
Denote P ∈ Ω K , where P is arranged such that
20
5
13
16
18
4
12
H0 (x, ξ ) =
15
2
7
and let
14
17
F:chipot506.tex; VTEX/Rita p. 76
(5.18)
42
43
44
Then D → +∞ is equivalent to β → 0.
45
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Existence and stability of spikes
1
2
3
4
5
6
563
The stationary system for (GM) is the following system of elliptic equations:
⎧
a2
2
⎪
⎨ a − a + h = 0, a > 0
h − β 2 h + β 2 a 2 = 0, h > 0
⎪
⎩ ∂a ∂h
∂ν = ∂ν = 0
1
2
3
in Ω,
in Ω,
on ∂Ω.
(5.19)
9
10
11
12
13
14
15
7
The following concerns the existence of symmetric K-peaked solutions in a twodimensional domain which generalizes the one-dimensional result Theorem 4.2.
T HEOREM 5.5. (See [87].) Let P0 = (P10 , P20 , . . . , PK0 ) be a non-degenerate critical point
of F0 (P) (defined by (5.16)). Moreover, we assume that the following technical condition
holds
18
19
20
21
22
(5.20)
→0
24
26
ξ =
27
28
29
⎧1
⎪
K
⎪
⎪
⎨
⎪
⎪
⎪
⎩
|Ω|
2 R2 w 2 (y) dy
1
|Ω|
η 2 2 w 2 (y) dy
R
1
|Ω|
K+η0 2 2 w 2 (y) dy
R
32
33
34
35
36
10
11
12
13
15
if η → ∞,
17
18
19
20
21
22
23
if η → 0,
24
(5.21)
if η → η0 ,
25
26
27
28
29
and
30
31
9
16
where η is defined by (5.18).
Then for sufficiently small and D = β12 sufficiently large, problem (5.19) has a solution
(a , h ) with the following properties:
x−Pj
(1) a (x) = ξ ( K
j =1 w( ) + O(k(, β))) uniformly for x ∈ Ω̄. Here w is the
unique solution of (2.8) and
23
25
8
14
if K > 1, then lim η = K,
16
17
5
6
7
8
4
30
k(, β) := ξ β .
2
2
(5.22)
31
32
(By (5.21), k(, β) = O(min{ 1 1 , β 2 }).)
log Furthermore, Pj → Pj0 as → 0 for j
33
34
= 1, . . . , K.
(2) h (x) = ξ (1 + O(k(, β))) uniformly for x ∈ Ω̄.
35
36
37
37
38
38
39
5.3. Existence of multiple asymmetric spots
39
40
41
42
40
Similar to the on dimensional case, there are also multiple asymmetric spots in a twodimensional domain. But the existence of such patterns is only restricted when
43
44
45
41
42
43
lim
→0
D
log
1
< +∞.
(5.23)
44
45
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1
2
3
4
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J. Wei
We first derive the algebraic equations for the heights (ξ,1 , . . . , ξ,K ).
For β > 0 let Gβ (x, ξ ) be the Green function given by
5
6
1
2
3
Gβ − β 2 Gβ
∂Gβ
∂ν = 0
+ δξ = 0 in Ω,
4
on ∂Ω.
(5.24)
6
7
8
7
Recall that β 2 =
and therefore β ∼
1
D
9
10
11
12
13
14
)1
log 1
. Let G0 (x, ξ ) be the Green function defined
β −2
+ G0 (x, ξ ) + O β 2
Gβ (x, ξ ) =
|Ω|
17
18
19
20
21
10
11
12
(5.25)
24
25
26
27
28
29
x − Pj
,
χ,Pj (x) = χ
δ
32
33
34
35
x ∈ Ω, j = 1, . . . , Km,
38
39
40
41
42
(5.26)
where χ is a smooth cut-off function which is equal to 1 in B1 (0) and equal to 0 in
B2 (0).
Let us assume the following ansatz for a multiple-spike solution (a , h ) of (GM):
R2
\
45
17
18
20
21
23
24
25
26
K
x−Pi
i=1 ξ,i w( )χ,Pi (x),
a ∼
h (Pi ) ∼ ξ,i ,
27
(5.27)
28
29
30
where w is the unique solution of (2.8), ξ,i , i = 1, . . . , K, are the heights of the peaks, to
be determined later, and P = (P1 , . . . , PK ) are the locations of K peaks.
Then we can make the following calculations, which can be made rigorous with error
terms of the order O( 1 1 ) in H 2 (Ω).
log
31
32
33
34
35
From the equation for h ,
36
h − β 2 h + β 2 a2 = 0,
h Pi =
37
38
39
we get, using (5.25),
43
44
16
22
36
37
14
19
30
31
13
15
in the operator norm of L2 Ω) → H 2 (Ω). (Note that the embedding of H 2 (Ω) into L∞ (Ω)
is compact.)
We define cut-off functions as follows: Let P ∈ Ω K . Introduce
22
23
8
9
in (5.14).
In Section 2 of [87] a relation between G0 and Gβ is derived as follows
15
16
5
40
Ω
Gβ Pi , ξ β 2 a2 (ξ ) dξ
=
Ω
β −2
|Ω|
2 2
+ G0 Pi , ξ + O β
β
41
42
43
44
45
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Existence and stability of spikes
1
×
2
K
2
ξ,j
w2
ξ − Pj
j =1
3
565
1
χ,Pj (ξ ) dξ
2
3
4
1
2
+ β G0 Pi , ξ + O β
=
Ω |Ω|
K
ξ − Pj
2
χ,Pj (ξ ) dξ.
×
ξ,j
w2
4
5
6
7
8
4
5
6
7
8
j =1
9
9
10
11
10
Thus
11
12
13
14
ξ,i
15
16
17
2 2 ξ − Pi
2 2
2 2
χ,Pi (ξ ) dξ
= ξ,i
w (y) dy + ξ,i β
G0 Pi , ξ w
|Ω| R2
Ω
1
2 2
2
+ β G0 Pi , Pj ξ,j w 2 (y) dy
+
|Ω|
R2
j =i
18
+
20
24
25
26
27
28
29
K
2 4
2
O β + O β 42 .
ξ,j
j =1
21
23
R2
= 2 G0 Pi , Pj
33
+
34
3
2
37
R2
∂Pj,l
= G0 Pi , Pj
36
19
(5.28)
20
R2
24
25
26
27
28
29
31
32
w (y)yl dy + O 4
2
R2
w (y) dy + O 4 .
2
38
33
34
35
36
37
38
(Note that we have set y =
ξ −Pj
and we have used the relation
42
R2
39
40
41
w 2 (y)yl dy = 0
44
45
17
30
w 2 (y) dy
K ∂G (P , P ) 0 i
j
l=1
35
43
16
23
2 ξ − Pj
χ,Pj (ξ ) dξ
G0 Pi , ξ w
Ω
G0 Pi , y + Pj w 2 (y) dy + e.s.t.
= 2
32
41
15
22
31
40
14
21
Here we have used that for j = i
30
39
13
18
19
22
12
42
43
44
which holds since w is radially symmetric.)
45
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1
J. Wei
Using (5.15) in (5.28) gives
2
3
4
F:chipot506.tex; VTEX/Rita p. 80
ξ,i
2
2 = ξ,i
|Ω|
5
1
2
R2
3
w 2 (y) dy
4
ξ − Pi
1
1
log − H0 Pi , ξ w 2
2π
|P
−
ξ
|
Ω
i
1
2 2
+
+ β 2 G0 Pi , Pj ξ,j
w 2 (y) dy
|Ω|
R2
2 2
+ ξ,i
β
6
7
8
9
5
χ
,Pi
(ξ ) dξ
7
8
9
j =i
10
11
+
12
13
K
2
ξ,j
j =1
14
15
=
16
2
ξ,i
2
|Ω|
10
11
2 4
O β + O β 42
2
R2
w (y) dy
2
+ ξ,i
12
13
β2 2
1
log
2π
14
15
2
R2
w (y) dy
16
2 1
2 β
w 2 (y) log
w 2 (y) dy
+ ξ,i
2
dy − 2 H0 Pi , Pi
2π
|y|
R2
R2
1
2 2
+
w 2 (y) dy
+ β 2 G0 Pi , Pj ξ,j
|Ω|
R2
17
18
19
20
21
j =i
22
23
+
24
K
(5.29)
28
ξ,i =
K
2
ξ,j
j =1
32
33
34
+
35
K
2
|Ω|
2
ξ,j
O
R2
2
w 2 (y) dy + ξ,i
β2
2π
2 log
1
R2
30
w 2 (y) dy
31
32
(5.30)
45
34
35
36
37
Let us rescale
38
ξ,i = ξ ξ̂,i ,
where ξ =
|Ω|
.
2 R2 w 2
ξ,i =
39
(5.31)
40
41
42
Then from (5.30) we get
43
44
24
33
2 2
β .
41
42
22
28
38
40
21
27
j =1
36
39
20
29
30
37
19
26
Recall that H0 ∈ C 2 (Ω̄ × Ω).
Considering only the leading terms in (5.29) we get following
29
31
18
25
26
27
17
23
2 4
2
O β + O β 42 .
ξ,j
j =1
25
6
η
1
+
ξ 2 2
|Ω| |Ω| ,i
43
R2
w 2 (y) dy
44
45
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Existence and stability of spikes
1
+
2
5
ξ̂,i → ξ̂i ,
7
9
2
|Ω|
R2
w 2 (y) dy +
K
2
ξ,j
O β 22 ,
1
2
j =1
3
4
where η was introduced in (5.18). Assuming that
6
8
2
ξ,j
j =i
3
4
567
η → η0 ,
5
(5.32)
ξ̂,i =
11
K
9
10
2
2
+ ξ̂,i
η0 ,
ξ̂,j
i = 1, . . . , K,
(5.33)
j =1
12
15
13
which can be determined completely.
In fact, let
16
14
15
16
ρ(t) = t − η0 t 2 .
17
(5.34)
18
19
21
ρ(ξ̂i ) =
22
K
19
20
ξ̂j2 ,
i = 1, . . . , K,
(5.35)
j =1
23
25
ρ(ξ̂i ) = ρ(ξ̂j )
for i = j.
(5.36)
27
30
31
32
28
(ξ̂i − ξ̂j ) 1 − η0 (ξ̂i + ξ̂j ) = 0.
37
38
39
40
41
42
43
44
45
29
(5.37)
ξ̂i − ξ̂j = 0 or
32
ξ̂i + ξ̂j =
1
.
η0
33
(5.38)
34
35
The case of symmetric solutions (ξ̂i = ξ̂1 , i = 2, . . . , N ) has been studied in [86] and
[87]. Let us now consider asymmetric solutions, i.e., we assume that there exists an
i ∈ {2, . . . , N} such that ξ̂i = ξ̂1 . Without loss of generality, let us assume that
36
37
38
39
40
ξ̂2 = ξ̂1 ,
41
42
which implies that
ξ̂1 + ξ̂2 =
30
31
Hence for i = j we have
35
36
26
27
That is
33
34
22
24
which implies that
26
29
21
23
25
28
17
18
Then (5.33) is equivalent to
20
24
11
12
13
14
7
8
we obtain the following system of algebraic equations
10
6
43
1
.
η0
44
(5.39)
45
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1
2
3
4
5
6
7
F:chipot506.tex; VTEX/Rita p. 82
J. Wei
Let us calculate ξ̂j , j = 3, . . . , K. If ξ̂j = ξ̂1 , then by (5.38), ξ̂j + ξ̂1 =
1
η0 ,
which implies
1
that ξ̂j = ξ̂2 .
Thus for j 3, we have either ξ̂j = ξ̂1 or ξ̂j = ξ̂2 .
Let k1 be the number of ξ̂1 ’s in {ξ̂1 , . . . , ξ̂K } and k2 be the number of ξ̂2 ’s in {ξ̂1 , . . . , ξ̂K }.
Then we have k1 1, k2 1, k1 + k2 = K.
This gives
2
8
9
10
13
14
15
16
ξ̂1 − η0 ξ̂12 =
K
ξ̂j2 = k1 ξ̂12 + k2 ξ̂22 ,
19
24
= k1 ξ̂12
+ k2
1
− ξ̂1
η0
16
2
17
18
19
20
and therefore
21
22
(k1 + k2 + η0 )ξ̂12
2k2 + η0
k2
−
ξ̂1 + 2 = 0.
η0
η0
(5.42)
29
30
33
34
35
27
k2
(2k2 + η0 )2
4 2 (k1 + k2 + η0 ).
η02
η0
28
(5.43)
38
39
40
41
42
43
44
45
29
30
31
The strict inequality of (5.43) is equivalent to
*
η0 > 2 k1 k2 .
32
33
(5.44)
36
37
24
26
Equation (5.42) has a solution if and only if
31
32
23
25
27
28
13
15
Substituting (5.41) into (5.40), we obtain
ξ̂1 − η0 ξ̂12
10
14
25
26
7
12
(5.41)
22
23
6
11
1
− ξ̂1 .
ξ̂2 =
η0
20
21
5
9
(5.40)
j =1
17
18
4
8
11
12
3
34
35
36
It is easy to see that if (5.44) holds, then there are two different solutions to (5.42) which
are given by (ρ± , η± ).
Therefore we arrive at the following conclusion.
√
L EMMA 5.6. Let η0 2 k1 k2 . Then the solutions of (5.33) are given by (ξ̂1 , . . . , ξ̂N ) ∈
({ρ± , η± })K√where the number of ρ± ’s is k1 and the number of η± ’s is k2 .
If η0 > 2√k1 k2 , there exist two solutions (ρ± , η± ).
If η0 = 2√k1 k2 , there exists one solution (ρ± , ρ± ).
If η0 < 2 k1 k2 , there are no solutions (ρ± , ρ± ).
37
38
39
40
41
42
43
44
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Existence and stability of spikes
1
2
3
4
5
6
7
8
9
10
11
√
Let η0 > 2 k1 k2 where k1 + k2 = K, k1 , k2 1. By Lemma 5.6, there are two solutions
to (5.33). In fact, we can solve
)
)
2k2 + η0 + η02 − 4k1 k2
2k2 + η0 − η02 − 4k1 k2
ρ+ =
,
ρ− =
,
(5.45)
2η0 (η0 + K)
2η0 (η0 + K)
)
)
2k1 + η0 − η02 − 4k1 k2
2k1 + η0 + η02 − 4k1 k2
,
η− =
.
(5.46)
η+ =
2η0 (η0 + K)
2η0 (η0 + K)
Note that
ρ+ + η+ =
13
14
16
17
21
22
26
For
P ∈ ΩK
40
41
42
43
44
45
9
10
13
17
(5.48)
18
19
20
21
22
23
24
for i = 1, 2, . . . , K.
27
28
29
K
H0 (Pk , Pk )ξ̂k4 −
k=1
31
G0 (Pi , Pj )ξ̂i2 ξ̂j2
(5.49)
i,j,=1,...,K, i=j
32
33
34
39
8
16
we define
F̂0 (P) =
33
38
7
30
32
37
6
26
31
36
5
15
30
35
4
25
Pi = (Pi,1 , Pi,2 )
28
3
14
with
27
29
(5.47)
P = (P1 , P2 , . . . , PK )
24
2
12
1
.
η0
Then there are k2 η’s in (ξ̂1 , . . . , ξ̂K ).
Let P = (P1 , . . . , PK ) ∈ Ω K , where P is arranged such that
23
25
ρ − + η− =
ξ̂j ∈ {ρ, η}, and the number of ρ’s in (ξ̂1 , . . . , ξ̂K ) is k1 .
19
20
1
,
η0
Let (ρ, η) = (ρ+ , η+ ) or (ρ, η) = (ρ− , η− ). We drop “±” if there is no confusion.
K be such that
Let (ξ̂1 , . . . , ξ̂K ) ∈ R+
18
1
11
12
15
569
34
and
35
M̂0 (P) = ∇P2 F̃0 (P).
(5.50)
Then we have the following theorem, which is on the existence of asymmetric K-peaked
solutions.
T HEOREM 5.7. (See [88].) Let K 2 be a positive integer. Let k1 , k2 1 be two integers
such that k1 + k2 = K. Let
√
|Ω|
1
β 2 |Ω|
β2 = ,
η =
log
,
D
2π
36
37
38
39
40
41
42
43
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J. Wei
√
where |Ω| denotes the area of Ω, Assume that η0 = lim→0 η > 2 k1 k2 ,
1
2
2
η0 = K
(T1)
3
3
4
5
4
and that
5
6
7
6
P0 = P10 , P20 , . . . , PK0 is a non-degenerate critical point of F̂0 (P)
(T2)
7
8
9
10
11
12
13
8
(defined by (5.49)).
Then for sufficiently small the stationary (GM) has a solution (a , h ) with the following properties:
x−Pj
1
(1) a (x) = K
j =1 ξ,j (w( ) + O( D )) uniformly for x ∈ Ω̄, where w is the unique
solution of (2.8) and
14
15
16
19
20
21
ξ,j = ξ ξ̂,j ,
|Ω|
ξ = 2 .
R2 w 2
Further, (ξ̂,1 , . . . , ξ̂,K ) → (ξ̂1 , . . . , ξ̂K ) which is given by (5.48).
(2) h (Pj ) = ξ,j (1 + D1 ) in H 2 (Ω), j = 1, . . . , K.
(3) Pj → Pj0 as → 0 for j = 1, . . . , K.
28
29
30
31
32
33
36
37
38
39
40
41
19
20
21
25
Next we study the stability and instability of the symmetric K-peaked solutions constructed
in Theorems 5.4 and 5.5.
In the strong coupling case, it turns out all solutions are stable:
T HEOREM 5.8. (See [86].) Suppose D = O(1). Let P0 and (a , h ) be defined as in Theorem 5.4. Then for and τ sufficiently small (a , h ) is stable if all eigenvalues of the matrix
MD (P0 ) = (∇P20 FD (P0 )) are negative. (a , h ) is unstable if one of the eigenvalues of the
matrix MD (P0 ) is positive.
26
27
28
29
30
31
32
33
34
In the weak coupling case, the stability of symmetric K-peaked solutions in a bounded
two-dimensional domain can be summarized as follows.
T HEOREM 5.9. (See [87].) Let P0 be a non-degenerate critical point of F0 (P) and for
sufficiently small and D = β12 sufficiently large let (a , h ) be the K-peaked solutions
constructed in Theorem 5.5 whose peaks approach
Assume (5.20) holds and further that
P0 .
35
36
37
38
39
40
41
42
(∗)
0
P is a non-degenerate local maximum point of F0 (P).
44
45
18
24
42
43
16
23
5.4. Stability of symmetric K-spots
34
35
13
22
25
27
12
17
23
26
11
15
(5.51)
22
24
10
14
17
18
9
43
44
Then we have
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Existence and stability of spikes
1
Table 1
1
2
3
4
5
6
7
8
571
K = 1, τ
K = 1, τ
K = 1, τ
K > 1, τ
K > 1, τ
K > 1, τ
Case 1
Case 2
Case 3 (η0 < K)
Case 3 (η0 > K)
stable
?
unstable
unstable
unstable
unstable
stable
stable
stable
stable
stable
stable
stable
?
unstable
unstable
unstable
unstable
stable
?
stable
stable
?
stable
3
small
finite
large
small
finite
large
9
14
15
2πD
|Ω|
log 1 ).
2πD
|Ω|
log 1 ).
24
25
26
27
30
33
20
C ASE 3. η → η0 ∈ (0, +∞) (i.e.,
2πD
|Ω|
∼
1
η0
log 1 ).
36
37
38
39
42
43
44
45
23
24
25
26
27
28
The statement of Theorem 5.9 is rather long. Let us therefore explain the results by the
following remarks.
29
30
31
R EMARK 5.4.1. Assuming that condition (∗) holds, then for small the stability behavior
of (a , h ) can be summarized in the following table:
32
33
34
R EMARK 5.4.2. The condition (∗) on the locations P0 arises in the study of small (o(1))
eigenvalues. For any bounded smooth domain Ω, the functional F0 (P), defined by (5.16),
always admits a global maximum at some P0 ∈ Ω K . The proof of this fact is similar to the
Appendix in [87]. We believe that in generic domains, this global maximum point P0 is
non-degenerate.
40
41
21
22
If K > 1 and η0 < K, then (a , h ) is linearly unstable for any τ > 0.
If η0 > K, then there exist 0 < τ2 τ3 such that (a , h ) is linearly stable for τ < τ2
and τ > τ3 .
If K = 1, η0 < 1, then there exist 0 < τ4 τ5 such that (a , h ) is linearly stable for
τ < τ4 and linearly unstable for τ > τ5 .
34
35
15
19
31
32
14
18
(a , h ) is linearly stable for any τ > 0.
28
29
13
17
22
23
8
16
C ASE 2. η → +∞ (i.e.,
20
21
7
12
If K = 1 then there exists a unique τ1 > 0 such that for τ < τ1 , (a , h ) is linearly stable,
while for τ > τ1 , (a , h ) is linearly unstable.
If K > 1, (a , h ) is linearly unstable for any τ 0.
18
19
6
11
16
17
5
10
C ASE 1. η → 0 (i.e.,
12
13
4
9
10
11
2
35
36
37
38
39
40
It is an interesting open question to numerically compute the critical points of F0 (P) and
link them explicitly to the geometry of the domain Ω.
We believe that for other types of critical points of F0 (P), such as saddle points, the
solution constructed in Theorem 5.5 should be linearly unstable. We are not able to prove
this at the moment, since the operator L is not self-adjoint.
41
42
43
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J. Wei
R EMARK 5.4.3. Case 1 and Case 3 with η0 < K resemble the shadow system and Case 2
and Case 3 with η0 > K are similar to the strong coupling case.
3
4
5
6
9
10
13
14
15
18
19
22
−
23
2[(1 + η0 (1 + τ λ0 ))
24
wφi + j =i
(K + η0 )(1 + τ λ0 ) R2 w 2
R2
R2
wφj ]
2η0
φ − φ + 2wφ −
(K + η0 ) R2 w 2
30
31
37
40
(5.52)
43
44
45
15
17
18
19
25
27
R2
28
w(y)φ(y) dy w 2 = λφ,
29
(5.53)
30
31
32
and
2(K + η0 (1 + τ λ0 )) R2 wφ 2
w = λ0 φ,
φ − φ + 2wφ −
(K + η0 )(1 + τ λ0 ) R2 w 2
φ ∈ H 2 R2 ,
33
34
35
36
(5.54)
37
38
where 0 < η0 < +∞ and 0 τ < +∞.
Problem (5.53) is the same as (3.7). For problem (5.54), we have the following result
41
42
14
24
38
39
13
23
By diagonalization, we obtain two NELPs:
29
36
12
26
28
35
10
22
w 2 = λ0 φi ,
26
34
9
21
i = 1, . . . , K.
25
33
8
20
φi − φi + 2wφi
21
32
6
16
The proof of Theorem 5.9 is again divided by two parts: large eigenvalues and small eigenvalues. For small eigenvalues, we relate them to the functional F (P). For large eigenvalues,
we obtain a system of NLEPs:
20
27
5
11
R EMARK 5.4.5. Roughly speaking, assuming that condition (∗) holds and that τ is small,
|Ω|
log 1 is the critical threshold for the asymptotic behavior
then for 1, DK () = 2πK
of the diffusion coefficient of the inhibitor which determines the stability of K-peaked
solutions.
16
17
4
7
R EMARK 5.4.4. We conjecture that in Case 3, τ2 = τ3 . This will imply that for any τ 0
and η0 > K, multiple spikes are stable, provided condition (∗) is satisfied. (It is possible
to obtain explicit values for τ2 and τ3 .)
11
12
2
3
From Case 2 and Case 3 of Theorem 5.9, we see that for multiple spikes (K > 1 ) large τ
may increase stability, provided that η0 > K. This is a new phenomenon in R 2 . It is known
that in R 1 , large τ implies linear instability for multiple spikes [8,34,59,60].
7
8
1
39
40
41
T HEOREM 5.10.
(1) If η0 < K, then for τ small problem (5.54) is stable while for τ large it is unstable.
(2) If η0 > K, then there exists 0 < τ2 τ3 such that problem (5.54) is stable for τ < τ2
or τ > τ3 .
42
43
44
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Existence and stability of spikes
1
2
3
4
573
P ROOF. Let us set
1
2
2(K + η0 (1 + τ λ))
.
f (τ λ) =
(K + η0 )(1 + τ λ)
(5.55)
4
5
6
5
We note that
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
lim f (τ λ) =
τ λ→+∞
R2
26
29
30
31
32
33
34
35
36
37
38
8
9
|∇φ|2 + φ 2 − 2wφ 2 + 2
R2
wφ
2
3 2w
− R 2 2
wφ 0,
( R2 w )
R2
25
28
7
2η0
=: f∞ .
K + η0
If η0 < K, then by Theorem 3.12(2), problem (3.52) with μ = f∞ has a positive eigenvalue
α1 . Now by perturbation arguments (similar to those in [8]), for τ large, problem (5.54)
has an eigenvalue near α1 > 0. This implies that for τ large, problem (5.54) is unstable.
Now we show that problem (5.54) has no non-zero eigenvalues with non-negative real
part, provided that either τ is small or η0 > K and τ is large. (It is immediately seen that
0
> 1 as τ λ → +∞ if η0 > K. Then Theorem
f (τ λ) → 2 as τ λ → 0 and f (τ λ) → η02η+K
3.12 should apply. The problem is that we do not have control on τ λ. Here we provide a
rigorous proof.)
We apply the following inequality (Lemma 3.8(1)): for any (real-valued function) φ ∈
Hr2 (R 2 ), we have
24
27
R2
R2
w2
w2 φ
= −λ0
40
44
45
12
13
14
15
16
17
18
19
20
22
23
25
26
27
28
29
30
(5.57)
2
R2 wφ
|φ| − f (τ λ0 )
w 2 φ̄.
2
R2
R2
R2 w
31
32
φ—and integrating over R 2 , we obtain
33
34
35
36
37
38
(5.58)
39
40
41
43
11
24
Multiplying (5.57) by φ̄—the conjugate function of
that
|∇φ|2 + |φ|2 − 2w|φ|2
R2
10
21
(5.56)
where equality holds if√and only if φ is a multiple
of w.
√
Now let λ0 = λR + −1λI , φ = φR + −1φI satisfy (5.54). Then we have
2 wφ
L0 φ − f (τ λ0 ) R 2 w 2 = λ0 φ.
R2 w
39
42
3
41
Multiplying (5.57) by w and integrating over
R2 ,
we obtain that
3
2w
2
w φ = λ0 + f (τ λ0 ) R 2
wφ.
R2
R2
R2 w
42
43
(5.59)
44
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2
3
4
5
6
7
8
9
J. Wei
Taking the conjugate of (5.59) we have
3
2w
2
R
¯
w φ̄ = λ0 + f (τ λ̄0 ) w φ̄.
2
R2
R2
R2 w
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
1
2
(5.60)
R2
= −λ0
3 |
2
2w
2 wφ|
2
R
R
|φ| − f (τ λ0 ) λ̄0 + f (τ λ̄0 ) .
2
2
R2
R2 w
R2 w
7
8
9
10
(5.61)
35
36
37
38
41
42
43
44
45
13
14
15
16
17
18
19
20
21
22
23
24
25
(5.62)
26
27
By the usual Pohozaev’s identity for (2.8) (multiplying (2.8) by y ·∇w(y) and integrating
by parts), we obtain that
3
w3 =
w2 .
(5.63)
2
2
2
R
R
28
29
30
31
32
33
Substituting (5.63) and the expression (5.55) for f (τ λ) into (5.62), we have
34
35
2
3 η0 + K + (η0 − K)τ λ + Re (η0 + K)(1 + τ λ̄0 ) (η0 + K)λ̄0
2
+ (η0 − K)τ |λ0 |2 0
36
37
38
39
40
11
12
33
34
4
6
We just need to consider the real part of (5.61). Now applying the inequality (5.56) and
using (5.60) we arrive at
3 R2 w
−λR Re f (τ λ0 ) λ̄0 + f (τ λ̄0 )
2
R2 w
3
3
2w
2w
− 2 Re λ̄0 + f (τ λ̄0 ) R 2 + R 2 ,
R2 w
R2 w
√
where we recall λ0 = λR + −1λI with λR , λI ∈ R.
Assuming that λR 0, then we have
3
2w f (τ λ0 ) − 12 + Re λ̄0 f (τ λ0 ) − 1 0.
R
2
R2 w
3
5
Substituting (5.60) into (5.58), we have that
|∇φ|2 + |φ|2 − 2w|φ|2
10
11
F:chipot506.tex; VTEX/Rita p. 88
39
which is equivalent to
3
(1 + μ0 τ λR )2 + λR + μ0 τ + τ + μ0 τ 2 |λ0 |2 λR
2
3 2 2
μ τ + μ0 τ − τ λ2I 0
+
2 0
40
41
42
43
44
(5.64)
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Existence and stability of spikes
1
2
575
where we have introduced μ0 := ηη00 −K
+K .
If η0 > K (i.e., μ0 > 0) and τ is large, then
1
2
3
3
3 2 2
μ τ + μ0 τ − τ 0.
2 0
4
5
(5.65)
6
7
8
9
λI
12
13
8
9
2 wφ
2
2
R
|φ| = Im −f (τ λ0 ) w φ̄ .
2
R2
R2
R2 w
10
11
12
13
14
Hence
15
17
7
11
16
15
+
2
|λI | f (τ λ0 ) R
R2
18
w4
w2
16
C
(5.66)
21
19
where C is independent of λ0 .
Substituting (5.66) into (5.64), we see that (5.64) cannot hold for λR 0, if τ is small. 22
27
24
25
Finally we study the stability or instability of the asymmetric K-peaked solutions constructed in Theorem 5.7.
28
29
30
33
34
35
36
37
T HEOREM 5.11. Let (a , h ) be the K-peaked solutions constructed in Theorem 5.7 for sufficiently small, whose peaks are located near P0 . Further assume that
(∗)
P is a non-degenerate local maximum point of F̂ (P).
*
34
35
36
37
38
2 k1 k2 < η0 < K
(5.67)
39
40
and
41
42
k1 > k 2 ,
(ρ, η) = (ρ+ , η+ ).
44
45
30
33
Then we have:
(a) (Stability)
Assume that
42
43
29
32
0
40
41
27
31
38
39
26
28
31
32
21
23
5.5. Stability of asymmetric K-spots
25
26
20
22
23
24
17
18
19
20
5
6
So (5.64) does not hold for λR 0.
To consider the case when τ is small, we have to obtain an upper bound for λI .
From (5.58), we have
10
14
4
43
44
Then, for τ small enough, (a , h ) is stable.
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J. Wei
(b) (Instability)
Assume that either
1
2
3
3
η0 > K
4
4
5
6
5
or
6
7
8
7
τ is large enough.
8
9
10
9
Then (a , h ) is linearly unstable.
10
11
12
13
11
A consequence of Theorem 5.11 is stable asymmetric patterns can exist in a twodimensional domain for a very narrow range of D, namely for
14
15
16
1
log
2πK
√
√
D
1
|Ω|
|Ω|
<
<
log
√
|Ω| 4π k1 k2
19
(5.68)
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
18
19
21
6. High-dimensional case: N 3
22
23
25
16
20
21
24
15
17
and small enough, where k1 and k2 are two integers satisfying k1 + k2 = K, k1 1, k2 1. In most cases, asymmetric patterns are unstable.
20
22
13
14
17
18
12
23
When N 3, there are very few results on the full Gierer–Meinhardt system. The difference between N 3 and N 2 lies on the behavior of the Green function: when N 2,
the Green function is locally constant (when N = 2, it is locally ∞). The limiting problem
is still a single equation (2.8). But when N 3, the Green function is like |x−y|1 N−2 . The
limiting problem when N 3 becomes
⎧
ap
⎪
⎨ a − a + hq = 0
r
h + ahs = 0
⎪
⎩
a, h > 0, a, h → 0
25
26
27
28
29
in
RN ,
in RN ,
as |y| → +∞.
30
(6.1)
31
32
33
Problem (6.1) seems out of reach at this moment. We believe that there should a radially
symmetric solution to (6.1) which is also stable.
As far as the author knows, the only result in higher-dimensional case is the existence of
radially symmetric layer solutions [62].
Let Ω = BR be a ball of radius R in RN . By scaling, we may take D = 1 and obtain
formally the following elliptic system
⎧
p
2 a − a + ahq = 0
⎪
⎪
⎪
⎨
m
h − h + ahs = 0
⎪
vsa > 0, h > 0
⎪
⎪
⎩ ∂a ∂a
∂ν = ∂ν = 0
24
34
35
36
37
38
39
40
41
in BR ,
in BR
in BR ,
on BR ,
42
(6.2)
43
44
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Existence and stability of spikes
1
577
where (p, q, m, s) satisfies
1
2
3
4
2
p > 1,
q > 0,
m > 0,
s 0,
qm
> 1.
(p − 1)(s + 1)
(6.3)
5
6
7
8
9
10
13
J (0) = 0,
J1 (0) = 1,
J1 > 0.
(6.4)
18
21
22
N −1 +
J2 − J2 + δ0 = 0,
r
J2 > 0,
J2 (+∞) = 0,
(6.5)
27
28
J1 (r) = c1 r
2−N
2
Iν (r),
J2 (r) = c2 r
2−N
2
Kν (r),
J1 =
31
32
33
sinh r
,
r
J2 (r) =
J2,R (r) = J2 (r) −
36
37
38
39
40
41
N −2
2
44
45
13
17
18
e−r
4πr
20
21
22
24
25
26
27
28
29
(6.7)
.
32
33
34
J2 (R)
J1 (r)
J1 (R)
(6.8)
35
36
37
and a Green function GR (r; r )
N −1 GR − GR + δr = 0,
GR +
r
30
31
38
39
GR (0; r ) = 0,
GR (R; r ) = 0.
(6.9)
40
41
42
43
12
23
(6.6)
Let w(y) be the unique solution for ODE 2.103. Let R > 0 be a fixed constant. We define
34
35
ν=
where c1 , c2 are two positive constants and Iν , Kν are modified Bessel functions of order ν.
In the case of N = 3, J1 , J2 can be computed explicitly:
29
30
10
19
where δ0 is the Dirac measure at 0.
The functions J1 (r) and J2 (r) can be written in terms of modified Bessel’s functions. In
fact
25
26
9
16
J2
23
24
8
15
19
20
7
14
The second function, called J2 (r), satisfies
16
17
6
11
N −1 J1 − J1 = 0,
J1 +
r
14
15
4
5
(The case of the whole RN is also included here, by taking R = +∞.)
Note that in (6.2), we have replaced a r by a m since we will use r = |x| to denote the
radial variable.
We first define two functions, to be used later: let J1 (r) be the radially symmetric solutions of the following problem
11
12
3
42
Note that
J2,R
(R) = 0,
43
lim J2,R (r) = J2 (r).
R→+∞
44
(6.10)
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J. Wei
For t ∈ (0, R), set
1
2
(N − 1)(p − 1) J1 (t) J2,R (t)
MR (t) :=
+
+
.
qt
J1 (t) J2,R (t)
3
(6.11)
5
6
7
5
When R = +∞, J2,+∞ (r) = J2 (r). We denote G+∞
M(t). That is,
8
9
10
G(r; r ) = c0 (r )N −1
(r; r )
as
G(r; r )
and M+∞ (t) as
13
14
15
8
J2 (r )J1 (r),
J1 (r )J2 (r),
for r < r ,
for r > r ,
9
(6.12)
18
12
(6.13)
15
Then we have the following existence result on layered solutions.
16
T HEOREM 6.1. (See [62].) Let N 2. Assume that there exist two radii 0 < r1 < r2 < R
such that
23
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25
26
27
28
29
30
31
32
33
34
35
36
MR (r1 )MR (r2 ) < 0.
(6.14)
Then for sufficiently small, problem (6.2) has a solution (a,R , h,R ) with the following
properties:
(1) a,R , h,R are radially symmetric,
q
p−1
(2) a,R (r) = ξ,R w( r−t
)(1 + o(1)),
(3) a,R (r) = ξ,R (GR (t ; t ))−1 GR (r; t )(1 + o(1)), where GR (r; t ) satisfies (6.9),
ξ,R is defined by the following
(1+s)(p−1)−qm
qm
m
ξ,R = w GR (t ; t )
41
42
43
44
45
20
R
22
23
24
25
26
27
28
29
30
(6.15)
31
32
and t ∈ (r1 , r2 ) satisfies lim→0 MR (t ) = 0.
33
34
It remains to check condition (6.14), which can be verified numerically. Under some
conditions on p, q, we can obtain the following corollary.
35
36
37
38
C OROLLARY 6.2. Assume that the following condition holds:
39
40
18
21
37
38
17
19
21
22
13
14
19
20
10
11
(N − 1)(p − 1) J1 (t) J2 (t)
M(t) :=
+
+
.
qt
J1 (t) J2 (t)
16
17
6
7
11
12
4
39
(N − 2)q
+ 1 < p < q + 1.
N −1
40
(6.16)
Then there exists an R0 > 0 such that for R > R0 and sufficiently small, problem (6.2)
i , hi ) concentrating on sphere {r = t } with
has two radially symmetric solutions (a,R
i
,R
MR (ti ) = 0, i = 1, 2, and 0 < t1 < t2 < R, i = 1, 2.
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3
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5
6
7
8
579
We remark that Corollary 6.2 is the first rigorous result on the existence to (6.2) of
positive solutions in dimension N 3. Next we consider the existence of bound states.
That is, we consider the following elliptic system in RN :
⎧
ap
2
N
⎪
⎨ a − a + hq = 0 in R ,
m
in RN ,
h − h + ahs = 0
⎪
⎩
a, h > 0, a, h → 0 as |x| → +∞.
13
14
6
(6.17)
9
10
We have the following result.
11
T HEOREM 6.3. (See [62].) Let N 2. Assume that there exist two radii 0 < r1 < r2 <
+∞ such that
19
20
21
22
23
24
25
26
27
28
29
(6.18)
Then for sufficiently small, problem (6.17) has a solution (a , h ) with the following
properties:
(1) a , h are radially symmetric,
q
p−1
(2) a (r) = ξ w( r−r
)(1 + o(1)),
(3) h (r) = ξ (G(r ; r ))−1 G(r; r )(1 + o(1)), where ξ is defined at the following
(1+s)(p−1)−qm
qm
m
ξ = w G(r ; r )
34
35
36
39
40
41
42
43
44
45
16
R
18
19
20
21
22
23
25
(6.19)
and r ∈ (r1 , r2 ) satisfying lim→0 M(r ) = 0.
26
27
28
29
30
Similarly we have the following corollary.
31
32
C OROLLARY 6.4. Assume that N 2 and that the condition (6.16) holds. Then for sufficiently small, problem (6.2) has a radially symmetric bound state solution (a , h )
which concentrates on a sphere {r = t0 } where M(t0 ) = 0.
37
38
14
24
32
33
13
17
30
31
12
15
M(r1 )M(r2 ) < 0.
17
18
7
8
15
16
3
5
11
12
2
4
9
10
1
33
34
35
36
37
By using the same method, it is not difficult to generalize the results of Theorem 6.1 to
other symmetric domains, such as annulus or the exterior of a ball. We omit the details.
Several interesting questions are left open. First, can multiple layered solutions to (6.2)
exist? Second, it would be an interesting question to study the stability of these “ring-like”
solutions. Numerical computations in two dimension indicate that the “ring-like” solutions
constructed in Theorem 6.1 are unstable and will break into several spots due to angular
fluctuations. Third, if we vary R from 0 to +∞, what is the relation between the layered
solution constructed in [52] for the single equation (2.4) and the solutions in Theorem 6.1?
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J. Wei
7. Conclusions and remarks
1
2
3
4
5
6
7
8
9
10
2
In this chapter, I have surveyed the most recent results on the study of Gierer–Meinhardt
system.
First, we consider the case D = +∞. In this case, the state-state problem becomes a singularly perturbed elliptic Neumann problem (2.4). Using the LEM, we established various
existence results on concentrating solutions. In particular, Theorem 2.5 gives a lower bound
on the number of solutions to (2.4). Several interesting questions are associated with (2.4).
First, is there a lower bound on the number of boundary spikes? What is the optimal bound
on the number of solutions to (2.4)? The followings are just some related conjectures
11
12
13
14
15
16
17
20
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
6
7
8
9
10
12
13
15
l(N −1)
16
17
number of boundary spikes to (2.4).
18
C ONJECTURE 2. Suppose the distance function d(P , ∂Ω) has l local maximum points.
Then there is at least
19
20
21
C
N l
22
23
24
number of interior spikes to (2.4).
25
C ONJECTURE 3. Suppose we have the energy bound J [u ] for some m N .
Assume that the concentration set Γ = {u > 12 } is connected. Then the limiting set Γ =
lim→0 Γ has Hausdorff dimension N − m.
26
Second, we consider the stability of spike solutions to the shadow system (2.2). By
studying both small and large eigenvalues, we have completely characterized the stability
(or instability) in the case of r = 2, 1 < p < 1 + N4 or r = p + 1. The study of the NLEP
(3.52) is not complete yet. Many interesting questions are still open: the case of general r,
the case of large τ , the uniqueness of Hopf bifurcation, etc. The non-linear metastability of
interior spike solutions is studied in [6]. The stability of boundary spikes is studied in [32],
through a formal approach. It can be proved that when D > D0 () 1, the full Gierer–
Meinhardt system converges to the shadow system [59,60,77,78]. However, the critical
threshold D0 () seems unknown.
Third, we consider the one- and two-dimensional Gierer–Meinhardt systems. For steady
states, we established the existence of symmetric and asymmetric K-peaked spikes. In 1D,
the bifurcation of asymmetric K-spikes occur when D < DK . In 2D, the bifurcation of
asymmetric K-spikes occur when D ∼ log 1 . We also obtain critical thresholds for the
stability of K-peaked solutions: If 1 there are stability thresholds
30
C m
44
45
5
14
C
21
22
4
11
C ONJECTURE 1. Suppose the mean curvature function H (P ) has l local minimum points.
Then there is at least
18
19
3
27
28
29
31
32
33
34
35
36
37
38
39
40
41
42
43
44
D1 () > D2 () > D3 () > · · · > DK () > · · ·
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Existence and stability of spikes
1
581
such that if
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
2
DK ()
lim
>1
→0
D
3
4
5
then the K-peaked solution is stable, and if
6
7
DK ()
<1
→0
D
lim
8
9
In 2D,
10
the critical threshold is 2πK . In 1D, the small eigenvalues determine the critical thresholds, while in 2D, the large eigenvalues give the critical thresholds. An interesting question is to obtain the next order term in the critical threshold for 2D (which should be
O(1) and location-dependent). The dynamics of multiple spikes in 1D and 2D is completely open. In 1D, the dynamical equation for the positions of the spikes is a system of
algebraic-differential-equations (ADE). A matched asymptotic analysis is given in [33]. In
2D, the dynamics of two well-separated spots is studied in [20] and it is shown that the two
spots will repel each other, provided that the initial distance between the two spots is large
enough. In a general two-dimensional domain, the dynamics of multiple spots should be
governed by ∇FD (P) or ∇F0 (P).
Finally, it is almost completely open as regards to three-dimensional Gierer–Meinhardt
system. The main difficulty is the study of the coupled system (6.1) which requires some
new insights. A layered bound state is constructed, but most likely it is unstable. An interesting question is to generalize Theorem 6.1 to general domains.
Although the analysis in this paper was carried out for the Gierer–Meinhardt system, the
results can certainly be generalized to a much wide class of non-local reaction diffusion
systems that have localized spike solutions.
12
then the K-peaked solution is unstable. In 1D, the critical threshold is DK ∼
√
log
|Ω|
1
.
K2
29
34
35
36
37
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45
17
18
19
20
21
22
23
24
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28
31
33
34
35
36
37
38
40
References
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44
16
39
40
43
15
32
I would like to thank Professor W.-M. Ni for introducing me to this subject and for
his hearty encouragements and many valuable discussions. I also like to thank many of
my collaborators for their patience and sharing with me their wonderful ideas: Professors H. Berestycki, S.-I. Ei, EN Dancer, M. del Pino, F.H. Lin, C. Gui, M. Kowalczyk,
T. Kolokolnikov, D. Iron, W.-M. Ni, M. Ward, M. Winter, S. Yan and many others. The
research is supported by an Earmarked Grant from RGC of HK.
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41
14
30
Acknowledgments
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Existence and stability of spikes
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