Mingfeng Zhao (mingfeng@math.ubc.ca)
Research Statement
My research interest is in nonlinear partial differential equations (PDEs). In particular,
I am interested in the existence, uniqueness and regularity issues of solutions to nonlinear
elliptic and parabolic partial differential equations, as well as long time behaviors of solutions.
My current research focuses on the fractional Hardy-Schrodinger operators with interior and
boundary singularities, and nonlocal minimal surfaces. I am also open to new search topics,
especially in other nonlinear PDE problems.
1. Traveling Wave Solutions involving Fractional Laplacians
1.1. The existence of the traveling wave solution.
For any s ∈ (0, 1), the fractional Laplacian (−∆)s is often defined by the use of the
Fourier transformation. For any u ∈ S(Rn ), the fractional Laplacian (−∆)s u of u is defined
\s u(ξ) = (2π|ξ|)2s û, that is, (−∆)s is a pseudo-differentiable opertor with the symbol
as (−∆)
(2π|ξ|)2s . It is well known (see [32]) that equivalently we have
Z
s22s Γ n+2s
u(x) − u(y)
s
2
(1)
(−∆) u(x) = n
P.V.
dy, ∀x ∈ Rn .
n+2s
π 2 Γ(1 − s)
Rn |x − y|
The above integral definition shows the nonlocal feature of (−∆)s u, and can be used to
define (−∆)s for more general functions, e.g., u ∈ C 2 (Rn ).
The fractional Laplacians in terms of (1) and the integro-diffeo operators, which could
be the nonlinear generalization of the fractional Laplacians, share many similar properties
with the usual Laplacian, e.g., the potential theory, regularity theory, maximum principle,
Harnack inequality, etc. Recently, there have been a lot of research about theses operators and
differential equations involving these operators (see [31, 46] and reference therein).
Front propagation is a natural phenomenon which has appeared in phase transition, chemical reaction, combustion, biological spreading, etc. The mechanism of front propagation is
often the competing effects of diffusion and reaction. Traveling wave solutions are typical
profiles of physical states near the propagating front, and are therefore of great importance in
the study of reaction diffusion processes. There has been a tremendous amount of literature
on traveling wave solutions in mathematics as well as in various branches of applied sciences
(see [4, 19, 49] and references therein). Traveling wave solutions are essential building blocks
in various phase field models and play an important role in pattern formation and phase separation (see [2, 13]). Nonlocal phase transition models and related traveling wave solutions
have been studied in [3, 14, 51] and references therein, where the kernels of convolution in the
nonlocal operators are bounded, and in [21] where the kernels are periodic.
In the study of front propagation, traditionally the diffusion process is quite standard and
normal, in the sense that the concerned particles or objects are engaged in a Brownian motion
with a uniformly changed random variable. The resulting diffusion effect on the physical state,
mathematically is represented by the Laplacian of this function. Therefore, the difference
of various reaction diffusion systems relies on the nonlinear reaction effect which varies in
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Mingfeng Zhao (mingfeng@math.ubc.ca)
combustion, chemical reaction, phase transition, biological pattern formation, etc. In general,
a typical reaction diffusion system is in the form of
ut − ∆u = f (u),
(2)
where f (u) is a nonlinear function. If the front of a solution u in large time propagates at a
constant speed, the solution is typically close to a profile depending on the distance away from
the traveling front. We, therefore, study traveling wave solutions of one spatial variable. We say
that u ∈ C 2 (R2 ), is a traveling wave solution to (2) if u has the special form u(t, x) = g(x − µt)
for all (x, t) ∈ R2 . The constant µ is called the speed of the traveling wave u, and the function g
is called the profile of the traveling wave. We are interested in traveling front where g connects
the two states, say −1 and 1. The sliding method implies that such g strictly increases. The
study of traveling wave solution is reduced to the study of solution to the following system:

00
0

 −g − µg = f (g), ∀x ∈ R,
g 0 (x) > 0, ∀x ∈ R,
(3)

 lim g(x) = ±1.
x→±∞
There are three interesting cases:
• Fisher-KPP or Monostable Model:
f 0 (−1) > 0,
f 0 (1) < 0,
and f (t) > 0,
∀t ∈ (−1, 1).
• Combustion Model: There exists some t0 ∈ (−1, 1) such that
f (t) = 0, ∀t ∈ (−1, t0 ),
f (t) > 0, ∀t ∈ [t0 , 1],
and f 0 (1) < 0.
• Bistable or the Allen-Cahn Model: There exists some t0 ∈ (−1, 1) such that
f (t) < 0, ∀t ∈ (−1, t0 ),
f (t) > 0, ∀t ∈ (t0 , 1),
and f 0 (±1) < 0.
For the combustion and Fisher-KPP models, 0 and 1 are usually used as the limits with
only 1 being stable (monostable model). The concerned states in the Allen-Cahn model are often represented by −1 and 1 with both states being stable (bistable model). In the Allen-Cahn
model, there is also another nodal point t0 of f in (−1, 1); this nodal point may represents an
unstable state which is not the concerned state, since otherwise the equation may be regarded
as a Fisher-KPP equation by restricting u in (t0 , 1).
Using phase plane analysis, one can show that the traveling wave solutions (g, µ) always
exist for the above three types of non-linearities. More precisely, for the combustion and
bistable models, there exists a unique pair (g, µ) as the traveling wave solution to (3). For
the Fisher-KPP model, there exists a maximal speed µ0 such that for any speed µ ≤ µ0 ,
there exists a traveling wave solution (g, µ) to (3). Moreover, for the bistable non-linearity,
the traveling wave solution to (3) decays exponentially at infinity, i.e., 1 − g(x) ∼ e−ν|x| as
x → ∞ and −1 + g(x) ∼ e−ν|x| as x → −∞ for some positive constant ν > 0. In addition to
the method of phase analysis, one can also use the sub-super solution method and variational
method to prove the same results (see [50, 34] and references therein).
– Page 2 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
There have been a fast growing number of studies on the front propagation of the reaction
diffusion system with an anomalous diffusion such as super diffusion, which plays an important
role in various physical, chemical, biological and geological processes. Mathematically, such a
super diffusion is related to the Lévy process and may be modeled by a fractional Laplacian
operator (−∆)s u with 0 < s < 1, and the reaction diffusion equation with respect to such
super diffusion can be given by ut + (−∆)s u = f (u). We are interested in traveling wave
solutions in one dimensional spatial variable. Namely, we consider solutions u(x, t) in the form
of u(x − µt) for some constant µ, we will study the following problem:

s
0

 (−∆) u − µu = f (u), ∀x ∈ R,
u0 (x) > 0, ∀x ∈ R,
(4)

 lim u(x) = ±1.
x→±∞
Traveling wave solutions for reaction diffusion equations with the fractional Laplacians
have been studied in the last few years (see [10, 12, 38, 50]). It is very interesting to see
that the study of (4) really depends on the non-linearity f and the parameter s. Indeed,
when f is a Fisher-KPP non-linearity, it is known that the front propagation speed could be
very fast depending on the initial values (see [5]). The Fisher-KPP equation with a fractional
Laplacian displays a very different behavior, due to the super diffusion process involved. It was
discovered numerically in [16, 15, 37] that the front propagation can accelerate exponentially
in time. This phenomenon is rigorously studied and proved in [8] that for any s ∈ (0, 1), the
position of all level sets of solution u to reaction diffusion equation ut + (−∆)s u = f (u) with
|u| ≤ 1 moves exponentially fast in time t. Since a traveling wave front propagates linearly in
t, it is an immediate consequence that there is no traveling wave solution for the Fisher-KPP
equation with a fractional Laplacian.
For the combustion model, the results are much more interesting. Via the sub-super
solution method, it is shown in [38] that when f is a combustive non-linearity, if s ∈ (1/2, 1),
there exists a unique pair (µ, u) as solution to (4), and the solution u decays algebraically at
−∞. On the other hand, by the s-harmonic extension, it is shown in [25] that if s ∈ (0, 1/2],
there does not exist a traveling wave solution to (4) for the combustion model. Moreover,
they showed the nonexistence of traveling fonts for more general non-linearities including the
Fisher-KPP case.
There are two types of bistable non-linearities: balanced and unbalanced. ZWhen the
u
bistable non-linearity is balanced, i.e., the associated double well potential G(u) = −
f (t) dt
−1
has two wells with equal depths G(1) = G(−1) = 0, a traveling wave solution with one spatial
variable for the balanced Allen-Cahn equation is indeed a standing wave, i.e., the speed µ must
be zero. Such a solution is sometimes called a layer solution as it describes a transition layer
structure near the interface between two physical states. The existence of the standing wave
solution to (4) was proved in [9, 11] as a stationary point of the functional
Z Z
Z
1 1 1−2s
s/2 2
2
Es (u) =
(−∆) u + G(u) dx =
y
|∇u| (x, y)dxdy + G(u(x))dx,
R 2
R2+ 2
R
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Mingfeng Zhao (mingfeng@math.ubc.ca)
where u is the s-harmonic extension of u. In the same papers, they also proved the standing
wave solution is unique and decays algebraically at ±∞.
For the unbalanced fractional Allen-Cahn equation (where G(−1) 6= G(1)), we proved the
following similar results:
Theorem 1 (C. Gui and M. Zhao [27]). Let s ∈ (0, 1) and f ∈ C 2 (R) be any bistable nonlinearity, there exists a unique traveling wave solution u and unique speed µ to (4). Moreover,
1
1
1 − u(x) ∼ 2s as x → +∞ and −1 + u(x) ∼ 2s as x → −∞.
|x|
|x|
Let me explain the main idea of the proof of Theorem 1. To prove the existence of a
traveling front to (4) in the bistable case for all s ∈ (0, 1), we use a continuation argument.
The continuation argument has also been used in the study of nonlocal problems in [3, 20],
where a family of operators is used to connect nonlocal operators to the classical elliptic
operators. To be more precise, we consider a family of double well potentials Gθ , θ ∈ [0, 1] with
Gθ (u) = (1 − θ)G0 (u) + θG(u), where G0 (u) is a double well potential with equal depths and
f0 (u) = −G00 (u) < 0. It is easy to see that Gθ also satisfies the bistable property for all θ ∈ [0, 1].
1
Without loss of generality, we may assume that t0 = 0 and we can choose G0 (u) = (1 − u2 )2
4
as in the original Allen-Cahn equation. Let fθ (u) = −G0θ (u) = (1 − θ)(u − u3 ) + θf (u) be the
corresponding bistable nonlinear reaction. We shall consider a family of fractional Allen-Cahn
equations:

 (−∆)s uθ − µθ u0θ − fθ (uθ ) = 0, x ∈ R,
(5)
 lim u(x) = ±1.
x→±∞
It is proved in [9, 11] that when θ = 0 (hence f0 is a balanced bistable non-linearity,
µ0 = 0), (5) has a standing wave solution. Based on the invertibility of the linearized equation
of (5) (we can think the traveling front is the first eigenvalue which is simple), using the
implicit function theorem we can find a local branch of traveling wave solutions (µθ , uθ ) in a
suitable function space for θ sufficiently small. Once we can show that the branch of solutions
can be extended to θ = 1, we obtain a traveling wave solution to (4) as desired. The key to
this continuation argument is to have a uniform bound of µθ for all possible traveling wave
solutions (µθ , uθ ).
For future research in this direction, I am interested in the existence of the traveling wave
solution involving more general nonlocal operators. In particular, I want to study the following
nonlocal operator
Z
[u(x + y) + u(x − y) − 2u(x)]K(y)
dy, ∀x ∈ R,
LK,s u(x) = −
|y|1+2s
R
where s ∈ (0, 1) and K is a positive smooth function on R. In particular, if K is a positive
constant function on R, then LK,s is the same as (−∆)s . The study of the traveling wave
– Page 4 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
solution involving LK,s is reduced to the study of the following system:


 LK,s u − µ = f (u), ∀x ∈ R,
u0 (x) > 0, ∀x ∈ R,
(6)

 lim u(x) = ±1.
x→±∞
It’s natural to ask the following question:
Problem A. What conditions should be posed on s, K and f such that there is a traveling
wave solution to (6)? When a solution exists, how about uniqueness and asymptotic behaviors
at ±∞?
1.2. Convergence of traveling wave solutions.
In the phase field theory involving fractional Laplacians (see [29]), for a given balanced
bistable non-linearity f , and any small > 0, let f (u) = f (u) + , then f is a unbalanced
bistable non-linearity with m+ () and m− () instead of 1 and −1. Theorem 1 shows that for
any > 0, there exists a unique speed µ and unique traveling front u connecting m− () and
m+ (), that is,

s
0

 (−∆) u − µ u = f (u ) + , ∀x ∈ R,
u0 (x) > 0, ∀x ∈ R,
(7)

 lim u (x) = m± ().
x→±∞
The dependence of µ on is quite important in the phase field theory. By a similar
2
method as [27], we can get a uniform estimates of speeds, and deduce that u → u0 in Cloc
(R)
µ
1
as → 0, moreover, there exists some constant C0 > 0 such that C0 ≤
≤
for all
C0
µ
converges up to a subsequence, as & 0. On the other hand,
> 0, which implies that
one can give some concrete examples in the classical Allan-Cahn equation to suggest that
µ is linearly dependent on , as & 0. Indeed, we can show that this is true in general.
µ
In order to get the limit of
, we use a Hamiltonian identity which holds for all s ∈ (0, 1]
Z
that µ |u0 |2 dx = G(m+ ()) − G(m− ()), which implies that it is equivalent to show that
R
Z
Z
0 2
lim |u | dx =
|u00 |2 dx, which is done in [26] by showing that up to translation, u0
&0
R
R
uniformly decay at a rate of
1
|x|1+2s
at ±∞. The following theorem gives the general version
to the above argument:
Theorem 2 (C. Gui and M. Zhao [26]). Let 0 < s ≤ 1, f be a balanced bistable non-linearity
and g be any smooth function on R. For any small > 0, let f (u) = f (u) + g(u), and µ be
the speed corresponding to the traveling front. Then there holds that
Z
Z
0 2
lim |u | dx =
|u00 |2 dx.
&0
R
R
– Page 5 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
Theorem 2 tells us how the speed depends on a linear perturbation of non-linearity f .
On the other hand, it’s natural to think that for any fixed bistable nonlinearity f , how does
the speed of the traveling waves for (−∆)s with f depend on the parameter s? In fact, we
have the following theorem:
Theorem 3 (C. Gui and M. Zhao [26]). Let f be a bistable nonlinearity and µs be the traveling
speed for f and s ∈ (0, 1), then µs → 0 as s & 0, and µs → µ1 as s % 1.
In Theorem 3, we know that µs → 0 as s & 0. A natural question is whether µs linearly
depends on s, that is,
µs
Problem B. Can we find a constant C which may depend on s and f such that
→ C as
s
s & 0?
2. Liouville Type Theorem for Higher Order Elliptic Systems
For any m ∈ N, p > 0 and q > 0, we are concerned with the question of the nonexistence
of positive solutions to the following system:

 (−∆)m u = v p ,
(8)
∀x ∈ RN .
(−∆)m v = uq ,

u > 0, v > 0,
When m = 1, (8) becomes the so called Lane-Emden system

 ∆u + v p = 0,
(9)
∀x ∈ RN .
∆v + uq = 0,

u > 0, v > 0
It has been conjectured that
Conjecture 1 (Lane-Emden Conjecture). For any N ∈ N and p, q > 0, (9) has positive
solutions if and only if p, q satisfy
1
1
N −2
+
≤
.
p+1 q+1
N
Conjecture 1 was completely solved if we assume solutions are positive radically symmetric (see [39, 43, 45]). To the best of our knowledge, the full conjecture has not been completely solved yet in a general setting. Several partial answers have been given over the years.
1
1
−2
Souto [48] proved the nonexistence of positive C 2 solutions below the curve p+1
+ q+1
= N
N −1
N +2
when p, q > 0. Felmer and de Figureiredo [18] showed that when 0 < p, q ≤ N
and
−2
N +2 N +2
2
(p, q) 6= N −2 , N −2 , (9) has no positive C solutions. Further evidence supporting the conjecture can be found in [40], where it is shown that there are no positive super-solutions to (9)
below the curve
1
1
2
1
1
p > 0, q > 0 :
+
=1−
max
,
.
p+1 q+1
N −2
p+1 q+1
– Page 6 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
We refer to the above curve as S-curve and the hyperbola
1
1
+ q+1
p+1
=
N −2
N
in the conjecture will
2(q+1)
,
≥
be referred as Sobolev’s hyperbola. For 0 < p, q, if pq ≤ 1 or pq > 1 and max 2(p+1)
pq−1
pq−1
N − 2, the nonexistence of positive solutions was proved by Serrin and Zou in [44]. Direct
calculation shows this is the same region of (p, q) as the region below and on the S-curve.
For the special case min (p, q) = 1, the conjecture was proved by C.-S. Lin [35] . Busca and
Manásevich [7] proved that if p, q > 0, pq > 1,
N −2
2 (p + 1) 2 (q + 1)
2 (p + 1) 2 (q + 1)
≤ min
,
≤ max
,
< N − 2,
2
pq − 1
pq − 1
pq − 1
pq − 1
and
2 (p + 1) 2 (q + 1)
N −2 N −2
,
6=
,
,
pq − 1
pq − 1
2
2
there are no positive classical solutions to (9). Recently, Conjecture 1 was fully solved in
the case N = 3 by Poláčik, Quittner and Souplet [41] and by Souplet [47] when N = 4.
Souplet
the conjecture when N ≥ 5 under the additional assumption that
[47] also proved
2(p+1) 2(q+1)
max pq−1 , pq−1 > N − 3.
Comparing to the Lane-Emden system, less is known about the higher order system (8).
1
1
+ q+1
> 1− 2m
,
It is proved in [28, 52] that if N > 2m, p, q ≥ 1 and (p, q) 6= (1, 1) satisfying p+1
N
then (8) has no nontrivial non-negative radial solutions. For general solutions, the results in
[28, 52] show that if p, q ≥ 1, (p, q) 6= (1, 1) satisfies
2m (p + 1) 2m (q + 1)
max
,
≥ N − 2m,
pq − 1
pq − 1
then (8) admits no positive solutions. It is also proved in [28] that (8) does not admit any
+2m
positive solutions if 1 < p, q < N
. Under the additional assumptions (−∆)i u > 0, (−∆)i v >
N −2m
0 for i = 1, 2 · · · , m − 1, Yan [52] proved that (8) admits no positive solutions if pq ≤ 1.
Using a Rellich-Pohozaev identity combined with an adapted idea of measure and feedback
argument in Souplet’s paper [47], we prove the following improved Liouville type theorem:
Theorem 4 (F. Arthur, X. Yan and M. Zhao [1]). For any m ≥ 1, N > 2m, p ≥ 1 and q ≥ 1,
if (p, q) 6= (1, 1) and p, q satisfy
1
2m
1
(10)
+
>1−
,
p+1 q+1
N
and
2m (p + 1) 2m (q + 1)
,
≥ N − 2m − 1,
(11)
max
pq − 1
pq − 1
then (8) has no nontrivial non-negative solutions of class C 2m RN .
Moreover, if N = 2m + 1 or 2m + 2, (8) admits no positive solutions if p ≥ 1, q ≥ 1,
(p, q) 6= (1, 1) satisfies (10) .
In future, I am interested in studying the following problem:
Problem C. Can we prove Theorem 4 without the condition (11)?
– Page 7 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
3. The fractional Hardy-Schrödinger operator with interior singularity
Let 0 < α < 2, consider the following fractional Hardy-Sobolev best constant in Rn :
R
R |u|2
α
2
4
n |(−∆) u| dx − γ Rn |x|α dx
R
(12)
µγ,s,α (RN ) :=
inf
,
R
∗2
α
∗
2α (s)
u∈H 2 (Rn )\{0}
|u|2α (s)
dx
Rn |x|s
where 0 < α < 2, 0 ≤ s < α, 2∗α (s) =
2(n−s)
,
n−α
Γ2 ( n+α )
4
which is the
and 0 ≤ γ < γH := 2α Γ2 ( n−α
)
4
best fractional Hardy constant on Rn . An explicit formula for the extremal of µγ,s,α (Rn ) has
not been found yet - in the spite of the Laplacian case. Hence, we will try to understand the
profile of such extremal as much as possible. For which parameters γ and s, the best constant
µγ,s,α (Rn ) is attained, this was done in a recent analysis by Ghoussoub-Shakerian [23], where
the needed information on the existence of the extremal is given:
Theorem 5 (Ghoussoub-Shakerian [23]). Suppose 0 < α < 2, 0 ≤ s < α < n, and γ < γH .
(1) If either s > 0 or s = 0 and γ ≥ 0, then µγ,s,α (Rn ) is attained.
(2) If s = 0 and γ < 0, then there are no extremals for µγ,s,α (Rn ).
(3) If either 0 < γ < γH , or 0 < s < α and γ = 0, then any non-negative minimizer for
µγ,s,α (Rn ) is positive, radially symmetric, radially decreasing, and approaching zero as
|x| → ∞.
Many asymptotic properties of the positive extremals of µ0,s,α (Rn ) were given by Y. Lei
[33], Lu and Zhu [36], and Yang and Yu [53]. Recently, Dipierro, Montoro, Peral and Sciunzi
[17] found the asymptotic behavior of the extremal for µγ,0,α (Rn ) (i.e. 0 < γ < γH and s = 0)
near the origin and at infinity. Following the idea in [17], we can find the asymptotic behavior
of the extremal of µγ,s,α (Rn ) at zero and infinity in general case, which is 0 < γ < γH and
0 < s < α, and prove the following theorem.
Theorem 6 (Ghoussoub-Shakerian-Zhao[24]). For any 0 < s < α < 2, n > α and 0 < γ < γH ,
α
let u ∈ H 2 (Rn ) be a positive extremal for µγ,s,α (Rn ). Then there exists some constant C > 1
which only depends on N , s and α such that
C −1
[|x|1−ηγ
+
|x|1+ηγ −s ]
n−α
2−s
C
≤ u(x) ≤
[|x|1−ηγ
+
n−α
|x|1+ηγ −s ] 2−s
,
in Rn \{0},
where
ηγ = 1 −
β− (γ)(2 − s)
,
n−α
in which β− (γ) is the unique solution of the equation Ψn,α (β) = γ in (0, n−α
) with
2
β+α
Γ n−β
αΓ
2
2
.
Ψn,α (β) := 2
β
Γ n−β−α
Γ
2
2
– Page 8 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
Remark 1. Note that when α = 2, i.e., the fractional Laplacian boils down to the classical
Laplacian, the best constant associated with the Hardy-Sobolev inequality defined as
R
R
2
|∇u|2 dx − γ Rn |u|
dx
|x|2
Rn
n
µγ,s (R ) :=
inf
,
2
R |u|2∗ (s)
u∈D1,2 (Rn )\{0}
2∗ (s)
dx
Rn |x|s
2
where s ∈ [0, 2), 2∗ (s) = 2(n−s)
and 0 < γ < γH := (n−2)
( the best classical Hardy constant
n−2
4
n−2
in Rn ). The extremals for µγ,s (Rn ) are given by the functions u(x) = − 2 U (x/) for > 0,
where
1
in Rn \{0},
U (x) = h
n−2
(2−s)σ− (γ)
(2−s)σ+ (γ) i 2−s
|x| n−2 + |x| n−2
and
r
(n − 2)2
n−2
±
− γ.
σ± (γ) =
2
4
Let Ω be a smooth bounded domain in Rn with 0 inside the interior, we consider the best
constant for the fractional Hardy-Sobolev inequality on the domain Ω, namely,
R
R |u|2
α
2
4 u| dx − γ
|(−∆)
dx
Ω
Ω |x|α
µγ,s,α (Ω) :=
inf
,
2
α
R |u|2∗α (s)
2∗
u∈H 2 (Ω)\{0}
α (s)
dx
Ω |x|s
then we can show that µγ,s,α (Ω) = µγ,s,α (Rn ) by translating, scaling and cutting off the extremals of µγ,s,α (Rn ). So we know that there is an extremal for µγ,s,α (Ω), then it is also an
extremal for µγ,s,α (Rn ), which is impossible since Ω is bounded. We therefore resort to a setting
popularized by Brezis-Nirenberg [6] to de-homogenize the problem by considering the following
boundary value problem:

2∗ (s)−1
 (−∆) α2 u − γ u = u α
+ λu2 , in Ω
α
s
|x|
|x|
(13)

u = 0,
in ∂Ω,
α
where 0 < λ < λ1 (Lγ,α ) is the first eigenvalue of the operator Lγ,α = (−∆) 2 − |x|γα with the
zero Dirichlet boundary condition. One then should consider the quantity
R
R |u|2
R
α
2
|(−∆) 4 u|2 dx − γ Ω |x|
α dx − λ Ω |u| dx
Ω
µγ,s,α,λ (Ω) :=
inf
,
R
∗2
α
∗
2α (s)
u∈H 2 (Ω)\{0}
|u|2α (s)
dx
Ω |x|s
and use the fact that compactness is restored as long as µγ,s,α,λ (Ω) < µγ,s,α (Rn ), see [6] for
more details.
In the spirit of [30] which dealt with the Laplacian case, we observe that the behavior of
problem (13) is deeply influenced by the value of the parameter γ. Roughly speaking, when
γ is sufficiently near to γH , then the problem (13) becomes critical in the sense of PucciSerrin ([42]). The idea of studying how critical behavior occurs on varying a parameter γ on
which an operator Lγ,α continuously depends goes back to [30], where it is shown that the
– Page 9 –
Mingfeng Zhao (mingfeng@math.ubc.ca)
operator Lγ,2 := −∆ − |x|γ 2 has critical behavior in dimensions n ≥ 4 if and only if γ is critical,
2
2
which in this case means (n−2)
− 1 < γ < (n−2)
. Moreover, in [30] an explanation, as a
4
4
general principle of the critical behavior, is stated in terms of summability of the fundamental
solution: the critical behavior occurs precisely when the (generalized) fundamental solution of
the linear operator belongs to L2loc (Rn ) . This is clear if we think thatR a linear perturbation
in the differential equation corresponds to a quadratic perturbation λ Ω |u|2 dx
R in2 the related
functional, and this perturbation may deploy its effects for any small λ only if Ω u dx diverges
when λ → 0; but when λ → 0, u concentrates near the fundamental solution, hence we need
the condition that the fundamental solution is not square summable near the origin to get
nontrivial solutions to the nonlinear problem for any small λ.
When α = 2, i.e., in the case of the standard Laplacian, it’s well known that µγ,s,2,λ (Ω) is
2
attained if 0 ∈ Ω, n ≥ 4, s ≥ 0 and 0 ≤ γ ≤ (n−2)
− 1 and 0 < λ < λ1 (Lγ,2 . Recently, it has
4
been shown by Yang-Yu [53] that there exists a positive extremal for µ0,s,λ (Rn ) when s ∈ [0, 2).
On the other hand, for γ ∈ (0, γH ) and s ∈ [0, 2), and we can prove the the following theorem.
Theorem 7 (Ghoussoub-Shakerian-Zhao[24]). Let Ω be a smooth bounded domain in Rn (n ≥
2α) such that 0 ∈ Ω, 0 ≤ s < α. If 0 ≤ γ ≤ γ1 < γH , then there is at least one nontrivial
extremal for µγ,s,α,λ (Ω) for any 0 < λ < λ1 (Lγ,α ), where γ1 is the unique solution to the
equation β+ (γ) = n2 .
Remark 2. Note that the parameter γ1 in Theorem 7 plays the same role as the parameter
(n−2)2
− 1 in the classical Laplacian case. Indeed, we also can show that γ is not critical for
4
the operator Lγ,α if and only if 0 < γ ≤ γ1 , and it is critical when γ1 < γ < γH .
In this direction, I would like to study for the rest threshold, that is, γ1 ≤ γ < γH and
also want to get the similar results for the fractional Laplace in which 0 ∈ ∂Ω (one can see [22]
for the boundary case in classical Laplace).
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