n+2
MULTI-BUMP SOLUTIONS OF −∆u = K(x)u n−2 ON
LATTICES IN Rn
YANYAN LI, JUNCHENG WEI, AND HAOYUAN XU
Abstract. We consider the following semi-linear elliptic equation
with critical exponent:
n+2
−∆u = K(x)u n−2 ,
u>0
in Rn ,
where n ≥ 3, K > 0 is periodic in (x1 , ..., xk ) with 1 ≤ k < n−2
2 .
Under some natural conditions on K near a critical point, we prove
the existence of multi-bump solutions where the centers of bumps can
be placed in some lattices in Rk , including infinite lattices. We also
show that for k ≥ n−2
2 , no such solutions exist.
1. Introduction
We consider the following semi-linear elliptic equation with critical exponent:
(1.1)
n+2
−∆u = K(x)u n−2 ,
u > 0 in Rn .
Associated with (1.1) is the following energy functional
Z
2n
1
n−2
2
I(u) = kuk −
K(x)(u+ ) n−2 , u ∈ D,
2
2n
+
where u = max(u, 0) and D is the Hilbert space defined
R as the completion
∞
n
of Cc (R ) with respect to the scalar product hu, vi = Rn ∇u · ∇v and k · k
denotes the norm of D. By the maximum principle, a non-zero critical
point of I(u) will give rise to a positive solution to equation (1.1).
When K ≡ 1, all solutions of (1.1) have been classified by CaffarelliGidas-Spruck [14] and are given by:
n−2
2
n−2
λ
,
σP,λ (x) = (n(n − 2)) 4
1 + λ2 |x − P |2
for any λ > 0 and P ∈ Rn . See also Obata [42] and Gidas-Ni-Nirenberg
[27] when u has some natural decay as |x| → ∞.
When K is positive and periodic, Li proved that (1.1) has infinitely
many multi-bump solutions for n ≥ 3 in [32, 33, 34] by gluing approximate
solutions into genuine solutions with masses concentrating near isolated
1
2
Y.Y. LI, J. WEI, AND H. XU
sets of maximum points of K. Similar results were obtained by Yan in
[51] if K(x) has a sequence of strict local maximum points tending to
infinity. When K is positive and periodic, Xu constructed in [52] multibump solutions with mass concentrating near critical points of K including
saddle points; see also [53]. When K(x) is a positive radial function with
a strict local maximum at |x| = r0 > 0 and satisfies
K(r) = K(r0 ) − c0 |r − r0 |m + O(|r − r0 |m+θ ),
for some constant c0 > 0, θ > 0 and m ∈ [2, n − 2) near |x| = r0 ,
Wei and Yan constructed in [49] solutions with a large number of bumps
concentrating near the sphere |x| = r0 for n ≥ 5.
In this paper, we construct multi-bump solutions of (1.1) near critical
points of K(x) and the bumps can be placed on arbitrarily many or even
infinitely many lower dimensional lattice points. Furthermore we show
that the dimensional restriction is optimal.
More precisely, we assume the following conditions on K(x):
(H1) 0 < inf Rn K ≤ supRn K < ∞;
(H2) K ∈ C 1 (Rn ), K is 1-periodic in its first k variables;
(H3) 0 is a critical point of K satisfying: there exists some real number
β ∈ (n − 2, n) such that near 0,
(1.2)
K(x) = K(0) +
n
X
i=1
Pn
ai |xi |β + R(x),
[β]−1,1
(up to [β] − 1
where ai 6= 0,
i=1 ai < 0, and R(y) is C
derivatives are Lipschitz functions, [β] denotes the integer part of
P
s
−β+s
= o(1) as y tends to
β) near 0 and satisfies [β]
s=0 |∇ R(y)||y|
s
0. Here and following, ∇ denotes all possible partial derivatives
of order s.
Condition (H3) was used by Li in [34] for equation (1.1). Without loss
of generality, we may assume K(0) = 1. For any integer m ≥ 1 and integer
k ∈ [1, n], we define k-dimensional lattice
(1.3)
Qm := { all the integer points in [0, m]k × {0} ⊂ Rn },
where 0 ∈ Rn−k . We call x = (x1 , ..., xn ) ∈ Rn an integer point if all
x1 , ..., xn are integers.
The main results of this paper can be summarized as follows.
Main Theorem: For n ≥ 5, 1 ≤ k < n−2
, there exists l0 > 1 such that
2
for all l ≥ l0 and all m ≥ 1 (m can be +∞). There exists a C 2 positive
solution uQlm with bumps centered close to the lattice set Qlm (defined in
MULTI-BUMP SOLUTIONS ON LATTICES
(1.3)). Furthermore, this is optimal, i.e., if k ≥
exists.
n−2
,
2
3
then no such solution
In the following we give more precise statements of the above theorem.
For any λ > 0, we define the transformation Sλ : u(x) → uλ (x) :=
λ
u( λx ). Then for a solution u of (1.1), uλ (x) satisfies
− n−2
2
(1.4)
n+2
−∆uλ (x) = Kλ (x)uλ (x) n−2 ,
x ∈ Rn .
uλ > 0,
Since K is 1-periodic in its first k variables, Kλ (x) := K( λx ) is λ-periodic
in its first k variables.
For any positive integer l, let
n−2
(1.5)
λ = l β−n+2 ,
where n − 2 < β < n. (Throughout this paper, l and λ will satisfy the
relation (1.5).)
We scale the lattice Qm as
Xl,m = {λlx|x ∈ Qm }.
(1.6)
(m+1)k
For convenience, we order the points in Xl,m as {X i }i=1
.
For i = 1, ..., (m + 1)k , let P i ∈ B 1 (X i ) and Λi > 0. We use notations
(m+1)k
P := {P i }i=1
2
(m+1)k
and Λ := {Λi }i=1
. Then
(m+1)k
X
Wm (x, P, Λ) =
σP i ,Λi (x)
i=1
is an approximate solution of (1.4) when l is much larger than maxi Λi .
When there is no confusion, we denote Wm (x, P, Λ) by Wm (x).
For a fixed lattice Xl,m and τ > 1, and for functions φ, f ∈ L∞ (Rn ), let
−1

(m+1)k
X
1
 |φ(y)|,
(1.7)
kφk∗ = sup γ(y)
n−2
i |) 2 +τ
y∈Rn
(1
+
|y
−
X
i=1
and
(1.8)
kf |∗∗ = sup γ(y)
y∈Rn
where

(m+1)k
X
i=1
1
(1 + |y − X i |)
(m+1)k
(1.9)
n+2
+τ
2
1 + |y − X i | τ −1
)
γ(y) = min 1, min (
i=1
λ
!
−1

.
|f (y)|,
4
Y.Y. LI, J. WEI, AND H. XU
The above weighted norms depend on τ and the lattice Xl,m , since τ
and Xl,m are chosen. When there is no confusion, we just denote them as
above.
Remark 1.1. Without the function γ(y), the above weighted norms are
used by Wei-Yan in [49]. Similar weighted norms can be found in [18,
46, 47]. The reason for introducing γ(y) in the definition of the weighted
norms is crucial in our proofs. We shall comment more on this later.
We now state the first result.
Theorem 1. For n ≥ 5, 1 ≤ k < n−2
and m ≥ 1, assume that K satisfies
2
],
conditions (H1), (H2) and (H3). Then there exists a τ0 (n, k) ∈ (k, n−2
2
such that for any τ satisfying k ≤ τ < τ0 , there exist positive constants C1 ,
C2 , C and an integer l0 depending only on K, n, β, τ , such that for any
n−2
integer l ≥ l0 and λ = l β−n+2 , equation (1.1) has a positive C 2 solution u
satisfying
n+2
kSλ u − Wm k∗ ≤ Cλτ − 2 ,
with C1 < Λi < C2 for all i and max1≤i≤(m+1)k |P i − X i | → 0 as l → ∞,
uniformly in m.
If we allow the estimates to depend on m (e.g., the size of l0 ), then
m-bump solutions have been constructed by Xu in [52] for every 1 ≤
k ≤ n under the same assumption on K, see also [53]. As mentioned
earlier, Li constructed in [32, 33, 34] such m-bump solutions near isolated
sets of maximum points. The ansatz used in [32, 33, 34] is a variational
method as in [16, 17] of Coti Zelati-Rabinowitz and [48] of Séré which
glues approximate solutions into genuine solutions. On the other hand,
the method used in [52, 53] is gluing via implicit function theorem (or
a nonlinear Lyapunov-Schmidt technique), the same as that in our proof
of Theorem 1. Such Lyapunov-Schmidt reduction methods have been
developed and used by many aithors. We shall make some comments at
the end of this section.
The novelty and the main difficulty in the proof of Theorem 1 is that
all the estimates are independent of m and the results are optimal (see
Theorem 3 below). Thus we may construct multi-bump solutions on an
infinite lattice by letting m → ∞ while keeping l fixed (see Theorem 2
below). The new m-independent estimates are obtained by using the new
weighted norm k · k∗ (defined at (1.7)) as compared to k · k of D used
in [52]. Roughly speaking, k · k norm adds up errors near each bump,
while k · k∗ norm measures maximum of errors near each bump. The
reason for introducing γ(y) is to localize the estimate and to obtain better
decay estimates near each bump. This is crucially needed when we deal
MULTI-BUMP SOLUTIONS ON LATTICES
5
with the higher dimensional lattice case in which 2 ≤ k ≤ τ . As we
mentioned earlier, Wei-Yan [49] used a similar norm in which γ(y) ≡ 1.
They required that the number τ must be 1 + η̄ with η̄ > 0 being small.
Thus the norms used in [49] are only suitable for concentration on one
dimensional set (like circles as in [49]). Our norms work for any higher
dimensional concentration as long as the dimension k < n−2
(which is
2
optimal). This is one of major technical advances in this paper.
Let
Zk := {all integer points in Rk × {0}}, where 0 ∈ Rn−k ,
and, for an integer i ∈ [0, k],
Rki × {0} := {(x1 , ..., xk , 0, ..., 0) ∈ Rn |x1 , ..., xi ≥ 0}.
We often write Rki × {0} as Rki when there is no confusion. Note also
that Rk0 = Rk . Consider infinite lattices
and their scaled versions
Y i ≡ Y k,i := Zk ∩ Rki
Xli = λlY i .
Define
Wli =
X
σP (X),Λ(X) ,
X∈Xli
where P (X) ∈ B 1 (X) and Λ(X) ∈ (C1 , C2 ).
2
and 0 ≤ i ≤ k, assume that K
Theorem 2. For n ≥ 5, 1 ≤ k < n−2
2
satisfies conditions (H1), (H2) and (H3). Let τ , C1 , C2 , C and l0 be as
n−2
in Theorem 1. Then for any integer l ≥ l0 and λ = l β−n+2 , equation (1.1)
has a positive C 2 solution u satisfying
kSλ u − Wli k∗ ≤ Cλτ −
with Λ(X) ∈ (C1 , C2) for all X ∈
uniformly in X.
Xli
n+2
2
,
and |P (X) − X| → 0 as l → ∞
Solutions u constructed in Theorem 2 have infinitely many bumps. Indeed, u is close to σP (X),Λ(X) near every lattice point X ∈ Xli .
Theorem 2 follows from Theorem 1 by a limiting argument as follows.
For l ≥ l0 , let Xli be an infinite lattice as in Theorem 2. By Theorem 1, we
have solutions um of (1.1) for all m. For each m, we can find xm ∈ Xl,m ,
such that Xl,m − xm is monotonically increasing in m and
i
∪∞
m=1 (Xl,m − xm ) = Xl .
6
Y.Y. LI, J. WEI, AND H. XU
Let
(Sλ ûm )(x) = (Sλ um )(x + xm ).
Then ûm satisfies the same equation as um due to the periodicity of K.
See section 3 for details.
Remark 1.2. Theorem 2 shows a new phenomena that infinite-bump solutions can be constructed for semilinear elliptic equations with critical
exponents. For subcritical exponent semilinear elliptic equations, infinitebump solutions were constructed by Coti Zelati and Rabinowitz in [16, 17]
and Séré in [48]. There are two main differences (and difficulties) between the subcritical exponent problem (treated in [16, 17]) and critical
exponent problem: first, there is an extra loss of compactness-the scaling
invariance parameter. Second, there is the difficulty of controlling the algebraic decaying in an infinite lattice setting. (In [16, 17], the decay rate is
exponential.) As far as we know, this paper seems to be the first in obtaining the existence of solutions for critical exponent problems with infinitely
many bumps.
In an unpublished note [36], Li showed that the conclusion of Theorem
1 and Theorem 2 are false if n ≥ 3 and k ≥ n − 2. More specifically, let
K be a positive C 1 function which is periodic in each variable, satisfying,
for some constant β > n − 2, (∗)β condition for some positive constants
L1 and L2 in Rn :
|∇Ki | ≤ L1 , in Rn ,
[β]−1,1
and, if β ≥ 2, that K ∈ Cloc
(Rn ),
β−s
|∇s Ki (y)| ≤ L2 |∇Ki (y)| β−1 ,
for all 2 ≤ s ≤ [β],
∀y ∈ Rn .
Then for n ≥ 3 and k ≥ n − 2, there is no C 2 solution of (1.1) satisfying,
for some R, ǫ > 0 and 0 ≤ i ≤ k,
(1.10)
inf sup u ≥ ǫ.
x∈Rki BR (x)
(∗)β condition was introduced in [34]. If a positive C 1 periodic function
K is of the form (1.2) near every critical point of K, then K satisfies (∗)β
in Rn . Also (∗)β1 implies (∗)β2 if β1 ≥ β2 . If a function K in (1.1) satisfies
(∗)β for some β > n − 2, then solutions in any bounded region can only
have isolated simple blow up points, see [34].
Our next theorem improves this result to cover k ≥
mal in view of Theorem 1 and Theorem 2.
n−2
,
2
which is opti-
MULTI-BUMP SOLUTIONS ON LATTICES
7
, let K be as above. Then there is no
Theorem 3. For n ≥ 3 and k ≥ n−2
2
2
C solution of (1.1) satisfying (1.10) for some R, ǫ > 0 and 0 ≤ i ≤ k.
Remark 1.3. The solutions constructed in Theorem 2 satisfy (1.10) for
some R, ǫ > 0 and 0 ≤ i ≤ k. Since the hypotheses on K in Theorem 3
are stronger than that in Theorem 2, the assumption k < n−2
in Theorem
2
2 is optimal.
We end the introduction with some remarks and history on the finite/infinite dimensional reduction method. The original finite dimensional Liapunov-Schmidt reduction method was first introduced in a seminal paper by Floer and Weinstein [24] in their construction of single bump
solutions to one dimensional nonlinear Schrodinger equations (Oh [43] generalized to high dimensional case). On the other hand, Bahri [5] and BahriCoron [6] developed the reduction method for critical exponent problems.
In the last fifteen years, there are renewed efforts in refining the finite
dimensional reduction method by many authors. When combined with
variational methods, this reduction becomes ”localized energy method”.
For subcritical exponent problems, we refer to Ambrosetti-Malchiodi [1],
Gui-Wei [25], Malchiodi [39], Li-Nirenberg [37], Lin-Ni-Wei [38], Ao-WeiZeng [2] and the references therein. The localized energy method in degenerate setting is done by Byeon-Tanaka [12, 13]. For critical exponents,
we refer to Bahri-Li-Rey [8], Del Pino-Felmer-Musso [18], Rey-Wei [46, 47]
and Wei-Yan [49] and the references therein. Many new features of the
finite dimensional reduction are found. Our current work contributes to
this part of reduction method and gives an optimal treatment for critical
exponent problems. In recent years, there are new interests in extending
the finite dimensional reduction method to treat high dimensional concentration phenomena. This is the infinite dimensional reduction method and
has become very useful in constructing high dimensional concentration solutions. For compact manifold case, we refer to Del Pino-Kowalczyk-Wei
[20, 21] and Pacard-Ritore [45], and for noncompact manifolds, we refer
to Del Pino-Kowalczyk-Pacard-Wei [22], Del Pino-Kowalczyk-Wei [19] and
the references therein. A notable application of this infinite dimensional
reduction method is the construction of counterexample to De Giorgi’ s
Conjecture in large dimensions by M. del Pino, M. Kowalczyk and Wei
[19].
Throughout the paper, we will use the superscript to stand for a sequence of points in Rn and the subscript to stand for numbers or the
coordinates of a point in Rn unless otherwise stated.
8
Y.Y. LI, J. WEI, AND H. XU
The paper is organized as follows. In section 2, we carry out the
Lyapunov-Schmidt reduction. In section 3, we solve the finite dimensional problems and prove Theorem 1 and Theorem 2. In section 4, we
prove Theorem 3. In Appendices, we prove some basic lemmas that will
be used throughout the paper.
Acknowledgments: The research of the first author is partially supported by an NSF grant, the research of the second author is partially
supported by a General Research Fund from RGC of Hong Kong.
2. Finite Dimensional Reduction
In this section, we perform a finite-dimensional reduction to the rescaled equation (1.4). Let λ and l be given at (1.5) and Kλ (x) = K( λx ).
Consider the following energy functional
Z
2n
n−2
1
2
Kλ (u+ ) n−2 ,
Iλ (u) = kuk −
2
2n
u ∈ D.
Then any nonzero critical point of Iλ gives rise to a solution to (1.4). For
X i ∈ Xl,m , let P i ∈ B 1 (X i ). For any positive constants C1 < C2 , let
2
Λi ∈ (C1 , C2 ) and denote σi = σP i ,Λi . For a small ρ > 0, let
X
Σ = { (1 + ǫi )σPi ,Λi |P i ∈ B 1 (X i ), Λi ∈ (C1 , C2 ), |ǫi | < ρ}.
2
i
Then Σ is a smooth (n + 2)(m + 1)k dimensional sub-manifold in D.
When l is large enough, according to Proposition 2 of Bahri-Coron [7],
every function in a small tubular neighborhood of Σ in D can be uniquely
parameterized as
X
u(x) =
(1 + ǫi )σi (x) + φ(x) := W̄m (x) + φ(x),
i
where φ(x) is the unique minimizer of
min ku − wk.
w∈Σ
In particular, φ ∈ E, a subspace of D defined as
E := φ ∈ D|hZi,j , φi = 0, hσi , φi = 0, i = 1, ..., (m + 1)k , j = 1, ..., n + 1 ,
with Zi,j defined as
Zi,j =
∂σP i ,Λi
∂P i j
for 1 ≤ j ≤ n and Zi,n+1 =
∂σP i ,Λi
,
∂Λi
MULTI-BUMP SOLUTIONS ON LATTICES
9
where P i j means the j-th coordinate of P i . The size of the tubular neighborhood is independent of l and m. Obviously, E depends on the choice
of P and Λ.
In the form u = W̄m + φ, Iλ (u) = Iλ (W̄m + φ) can be written as
(2.1)
Z
Z
2n
1
n−2
2
J(P, ǫ, Λ, φ) =
|∇(W̄m + φ)| −
Kλ (y) (W̄m + φ)+ n−2 ,
2 Rn
2n Rn
where W̄m ∈ Σ and φ ∈ E.
By definition, u = W̄m + φ is a critical point of Iλ if and only if
(2.2)
∂J
(P, ǫ, Λ, φ) = 0,
∂φ
(2.3)
∂J
(P, ǫ, Λ, φ) = 0,
∂ǫ
(2.4)
∂J
(P, ǫ, Λ, φ) = 0,
∂P
(2.5)
∂J
(P, ǫ, Λ, φ) = 0.
∂Λ
We will use Lyapunov-Schmidt reduction method to solve these equations. More specifically, for fixed P and Λ, we first solve (2.2) and (2.3),
finding solutions φ(P, Λ) and ǫ(P, Λ) which are C 1 in P and Λ. Then
we use the Brouwer fixed point theorem to solve the finite dimensional
problems (2.4) and (2.5).
Let
(2.6)
(2.7)
F (φ, ǫ) :=
Gi (φ, ǫ) :=
∂J
(P, ǫ, Λ, φ);
∂φ
∂J
(P, ǫ, Λ, φ), i = 1, ..., (m + 1)k .
∂ǫi
The explicit expressions for F and G are as follows
Z
X
n+2
Gi (φ, ǫ) =
(1 + ǫj )hσi , σj i −
Kλ (W̄m + φ)+ n−2 σi ,
j
Rn
F (φ, ǫ) = φ − PE (−∆)−1 Kλ (W̄m + φ)+
where PE is the orthogonal projection of D onto E.
n+2
n−2
,
10
Y.Y. LI, J. WEI, AND H. XU
For fixed (P, Λ), setting
(2.8)
k
k
N(φ, ǫ) = (F, G)(φ, ǫ) : E × R(m+1) → E × R(m+1)
where G = (G1 , ..., G(m+1)k ). With the aid of the implicit function theorem, we solve
(2.9)
N(φ, ǫ) = 0.
We shall, as in [37], make use of the following form of the implicit function
theorem:
Lemma 2.1. (Brezis and Nirenberg) Let X , Y be Banach spaces, a > 0,
Ba = Ba (z0 ) = {z ∈ X : kz − z0 k ≤ a}. Suppose that F is a C 1 map of
Ba into Y,with F ′ (z0 ) invertible, and satisfying, for some 0 < θ < 1,
kF ′ (z0 )−1 F (z0 )k ≤ (1 − θ)a,
kF ′(z0 )−1 kkF ′ (z) − F ′ (z0 )k ≤ θ
∀z ∈ Ba .
Then there is a unique solution in Ba of F (z) = 0.
Define
M = {u ∈ L∞ (Rn )| kuk∗ < ∞},
D̃ = {u ∈ L∞ (Rn )| kuk∗∗ < ∞}.
We will solve the equation N(φ, ǫ) = 0 under the weak sense in Banach
spaces with norms related to k · k∗ , defined at (1.7). Since σi , Zi,j ∈ M,
we define a subspace of M by
n+2
4
R
R
n−2
M̃ := {φ ∈ M| Rn φσin−2 = 0,
φσ
Zi,j = 0,
n
i
R
(2.10)
for all i = 1, ..., (m + 1)k , j = 1, ..., n + 1}.
R
For functions φ ∈ M, hφ, σi i (or hσi , φi) should be understood as − Rn φ∆σi
(Similar statement also works for Zi,j ). Without introducing new symbols,
we still use PE to denote the orthogonal projection from M → M̃, which
can be defined as follows.
Let {f1 , ..., f(n+2)(m+1)k } be an orthonormal basis of the span{{σi }, {Zi,j }}
obtained by Gram-Schmidt procedure. Then, for φ ∈ D,
P(n+2)(m+1)k
hφ, fi ifi
PE φ = φ − i=1
= φ+
P(n+2)(m+1)k R
i=1
Rn
φ∆fi fi .
MULTI-BUMP SOLUTIONS ON LATTICES
Therefore, for any φ ∈ M, we define
X Z
PE φ = φ +
i
11
φ∆fi fi .
Rn
Then PE φ ∈ M̃ for every φ ∈ M. It is clear that if φ ∈ D ∩ M, then
the definitions are the same.
For any h ∈ D̃, we define (−∆)−1 h as
(2.11)
1
(−∆) h :=
n(n − 2)ωn
−1
Z
Rn
h(z)
dz
|y − z|n−2
where ωn is the volume of a unit ball in Rn . It is not difficult to see that
(2.11) makes sense. Lemma A.7 in appendix A shows that (−∆)−1 h ∈ M
and the proof of Lemma A.4 actually shows that PE (−∆)−1 is a well
defined map from M → M̃. Therefore, we can view N(φ, ǫ) as a map
k
k
from M̃ × R(m+1) → M̃ × R(m+1) . When (φ, ǫ) solves N(φ, ǫ) = 0 in
k
M̃ × R(m+1) , then (φ, ǫ) is automatically a solution to (2.8) in the weak
sense in the original spaces.
We will apply Lemma 2.1 to N at (0, 0) in the Banach spaces X = Y =
k
M̃ × R(m+1) , with norm
k(φ, ǫ)k = max(kφk∗ , λτ −1 |ǫ|),
where |ǫ| = maxi |ǫi |.
We have the following proposition.
Proposition 2.1. Under the assumptions of Theorem 1, when l ≥ l0 ,
k
equation (2.8) has a unique solution φ(P, Λ), ǫ(P, Λ) in M̃ × R(m+1) ,
with
C
k(φ, ǫ)k ≤ n+2 .
λ 2 −τ
Furthermore φ(P, Λ) and ǫ(P, Λ) are C 1 in P and Λ.
Proof. The proof will be carried out in several steps.
Step 1. We first show that, for some constant C > 0, independent of m
and l, we have the following crucial estimate for the error
(2.12)
kN(0, 0)k ≤
C
λ
n+2
−τ
2
.
Observe that
n+2
F (0, 0) = −PE (−∆)−1 Kλ Wmn−2 .
12
Y.Y. LI, J. WEI, AND H. XU
In view of Lemma A.7 and the fact that hF (0, 0), σi i = 0 for all i, we
just need to estimate klm k∗∗ , where the error becomes
n+2
X n+2
σin−2 .
(2.13)
lm = Kλ Wmn−2 −
i
Denote Ωi = {y ∈ Rn , |y − X i | ≤ |y − X j |, for all j 6= i}, Bi =
Bλl (X i ), and Bi,m = Bmax( m4 ,1)λl (X i ). Certainly Rn = ∪i Ωi .
For y ∈ Ωi , it holds that
n+2
n+2
n+2
X n+2
4
X n+2
y
y
n−2
n−2
n−2
n−2
n−2
σi | ≤ C|K( )−1|σi +C Ŵm,i + σi Ŵm,i +C
σjn−2 ,
|K( )Wm −
λ
λ
i
j6=i
where Ŵm,i =
P
j6=i
σj . We apply Lemma A.3 to estimate each term on
n+2
n−2
the right hand side of the above inequality. First we estimate Ŵm,i
.
By Lemma A.3, if y ∈ Ωi ∩ Bic ∩ Bi,m , we have
n+2
n−2
Ŵm,i
≤
≤C
C
4 k
(λl) n−2
P
j
P
j
1
4 k
n+2− n−2
(1+|y−xj |)
1
1
(1+|y−xj |)
n+2 +τ
2
For y ∈ Ωi ∩ Bi we obtain
P

C
n+2 −τ

j

(λl) 2

n+2
C
n−2
Ŵm,i ≤
≤
τ −1 P
(λl)n+2 
i

 1+|y−x |
j
λ
(λl)
n+2 −τ
2
1
(1+|y−xj |)
.
n+2 +τ
2
1
(1+|y−xj |)
n+2 +τ
2
,
y ∈ Ωi ∩ Bλ (xi )c
λτ −1
n
(λl) 2
,
y ∈ Bλ (xi ) ∩ Ωi .
c
If y ∈ Bi,m
∩ Ωi , by Lemma A.3, we have first
n+2
n−2
Ŵm,i ≤
(2.14)
≤C
X
j
1
(1 + |y − xj |)n−2
n+2
! n−2
n+2
1
m n−2 k .
i
n+2
(1 + |y − x |)
On the other hand
X
1
1
m k
(2.15)
≥
C
(1
+
C[
]) .
j |)n
i |)n
(1
+
|y
−
x
(1
+
|y
−
x
2
j
MULTI-BUMP SOLUTIONS ON LATTICES
13
c
It is not hard to see that when y ∈ Bi,m
∩ Ωi , there is a constant C
n−2
independent of m such that for 2 > k,
(2.16)
n+2
m
1
1
(1 + C[ ])k .
m n−2 k ≤ C
4
i
2
k
(1 + |y − x |)
2
(λl) n−2
c
From (2.14), (2.15) and (2.16), we conclude that for y ∈ Bi,m
∩ Ωi
n+2
X
1
1
n−2
≤C
.
Ŵm,i
n+2
4
+τ
j
2
(λl) n−2 k
j (1 + |y − x |)
Combining the previous estimates together, we get
n+2
C
n−2
k∗∗ ≤
.
(2.17)
kŴm,i
n+2
(λl) 2 −τ
Similarly, we have
4
kσin−2 Ŵm,i k∗∗ ,
k
C
n+2
X
σin−2 k∗∗ ≤
j6=i
(λl)
n+2
−τ
2
.
Now if y ∈ Ωi such that |y − xi | ≤ λ, we get
n+2
|K( λy ) − 1|σin−2 ≤
≤
C
n+2
λ 2 −τ
γ(y)
C|y−xi |β
1
(1+|y−xi |)n+2
λβ
P
j
1
(1+|y−xj |)
n+2 +τ
2
.
On the other hand, if y ∈ Ωi such that |y − xi | ≥ λ,
n+2
C
1
y
C X
.
|K( ) − 1|σin−2 ≤
≤
n+2
n+2
i
n+2
−τ
+τ
j
λ
(1 + |y − x |)
2
λ 2
j (1 + |y − x |)
n+2
Thus we obtain the estimate for k(Kλ − 1)σin−2 k∗∗ .
Combining the above inequalities, we get
C
klm k∗∗ ≤ n+2 −τ ,
λ 2
and therefore by Lemma A.7,
C
kF (0, 0)k∗ ≤ n+2 −τ .
λ 2
Next we estimate G(0, 0). For each i,
Z
X
Gi (0, 0) =
hσi , σj i −
j
Rn
n+2
Kλ Wmn−2 σi .
14
Y.Y. LI, J. WEI, AND H. XU
It is easy to see that
4
n+2
n+2
n+2
n−2
,
Wmn−2 = σin−2 + Cσin−2 Ŵm,i + C Ŵm,i
where C is some bounded constant which may vary and doesn’t depend
on m. Using Lemma A.3, Lemma A.1 and Lemma A.2, integrating in Ωj
and combining them together, we deduce that for n ≥ 5
Z
Z
n+2
n+2
C
C
n−2
n−2
Ŵm,i σi ,
≤ n.
σi ≤
Ŵm,i
n−2
(λl)
λ2
Rn
Rn
Similarly, we obtain that for n ≥ 5
Z
2n
C
C
|hσi , σi i −
Kλ σin−2 | ≤ β ≤ n ,
λ
λ2
Rn
|
X
X
hσj , σi i| ≤
j6=i
j6=i
|X j
C
C
≤ n.
i
n−2
−X |
λ2
Therefore we get that for each i,
|Gi (0, 0)| ≤
C
n .
λ2
Combining the estimates for F (0, 0) and Gi (0, 0) together, we obtain
(2.12).
Step 2. Next, we show that, for l large,
kN ′ (0, 0)−1 k ≤ C.
(2.18)
We consider the equation
(2.19)
N ′ (0, 0)(φ̃, ǫ̃) = (v, η).
For the v component, the equation becomes
(2.20)
4
4
X
n+2
n+2
PE (−∆)−1 Kλ Wmn−2 φ̃ −
PE (−∆)−1 Kλ Wmn−2 (
ǫ̃i σi ).
v = φ̃ −
n−2
n−2
i
For each i-th component, we have
(2.21)
Z
Z
4
4
X n+2
n+2
n−2
n−2
Kλ Wm σi φ̃+
Kλ Wm σi σj .
ηi = −
ǫ˜j hσi , σj i −
n − 2 Rn
n
−
2
n
R
j
MULTI-BUMP SOLUTIONS ON LATTICES
15
4
Since hv, σi i = 0, hφ̃, σi i = 0 for all i, in (2.20), we can replace Kλ Wmn−2 (
n+2
4
P
P
ǫ˜i σi ) −
ǫ̃i σ n−2 . Then we get
by l˜m := Kλ Wmn−2 (
i
φ̃(y) = v(y) +
(2.22)
i
i
n+2
n(n−2)2 ωn
1
+ n(n−2)ω
n
R
4
R
Rn
1
K (z)Wmn−2 (z)φ̃(z)dz
|z−y|n−2 λ
l̃m (z)
dz
Rn |y−z|n−2
= I + II + v +
P
P
i,j
+
P
i,j
ci,j Zi,j (y) +
ci,j Zi,j (y) +
For II, similar to the estimate of lm , we have
C|ǫ̃|
k˜lm k∗∗ ≤ n+2 −τ .
λ 2
P
P
i bi σi (y)
i bi σi (y).
4
To estimate ci,j , we multiply σsn−2 Zs,t on both side of (2.22). Modifying
the proof of Lemma A.7, we infer that
1
C
.
|ci,j |, |bi | ≤ Ck˜lm k∗∗ + n kφ̃k∗
λτ −1
λ2
Applying Lemma A.8, we obtain that, for l large,
|ǫ̃|
(2.23)
kφ̃k∗ ≤ C(kvk∗ + n+2 −τ ).
λ 2
To estimate (2.21), from Lemma A.6, we have
Z
4
Ckφ̃k∗
|
Kλ Wmn−2 σi φ̃| ≤ n−2 .
λ 2 +τ
Rn
For j 6= i, using Lemma A.1, we obtain
Z
4
n+2
C
|hσi , σj i −
Kλ Wmn−2 σj σi | ≤
n−2 ,
n − 2 Rn
|X i − X j | 2
and it is easy to get that for l large but independent of m,
Z
4
2
n+2
Wmn−2 σi2 ≤ −
hσi , σi i ≤ −C.
hσi , σi i −
n − 2 Rn
n−2
Therefore, we obtain that
!
kφ̃k∗
|ǫ̃|
(2.24)
|ǫ̃i | ≤ C |η| + n−2 +τ +
.
n−2
λ 2
(λl) 2
Estimates (2.23) and (2.24) yield
(2.25)
k(φ̃, ǫ̃)k ≤ C
k(v, η)k +
k(φ̃, ǫ̃)k
λ
n−2
2
!
,
i ǫ̃i σi )
16
Y.Y. LI, J. WEI, AND H. XU
therefore (2.18) follows when l is chosen large (independent of m).
Step 3. Next we estimate N ′ (φ, ǫ) − N ′ (0, 0) when k(φ, ǫ)k ≤
compute the term
1
.
2
We
(N ′ (φ, ǫ) − N ′ (0, 0))(v, η) = (ṽ, η̃).
For the ṽ component, it holds
(2.26)
!
X
4
4
n+2
ηi σi ){ (W̄m + φ)+ n−2 − Wmn−2 } ,
PE (−∆)−1 Kλ (v +
ṽ = −
n−2
i
and for the i-th component, we have
Z
4
X
4
n+2
(2.27) η̃i = −
Kλ σi (v +
ηj σj ){ (W̄m + φ)+ n−2 − Wmn−2 }.
n − 2 Rn
j
We first consider the term
kKλ v{ (W̄m + φ)+
Set ǫWm =
P
i ǫi σi
Then we get
(2.28) (W̄m + φ)+
4
n−2
4
− Wmn−2 }k∗∗ .
and Ω+ := {x ∈ Rn |u(x) ≥ 0} and Ω− = Ωc+ . Define

 1 if x ∈ Ω−
χΩ− (x) =

0 if x ∈ Ω+ .
4
n−2
4
4
4
4
−Wmn−2 = (W̄m +φ) n−2 −Wmn−2 +χΩ− |W̄m +φ| n−2 .
Since ǫ is small, there holds
(2.29)

4
−1
n−2


|ǫWm + φ|, if Wm ≥ |φ|,
W
m
4
4
n−2
|(W̄m + φ) n−2 − Wm | ≤ C

4

|φ| n−2 , if |φ| ≥ Wm ,
|φ|
4
n−2
4
n−2
|v| ≤ kφk∗
kvk∗
X
j
γ(y)
(1 + |y − xj |)
n−2
+τ
2
n+2
! n−2
.
Similar to the estimate of klm k∗∗ , we use Lemma A.3. If y ∈ Bi ∩ Ωi ,
n+2
! n−2
X
X
γ(y)
γ(y)
≤
C
.
n−2
n+2
j |) 2 +τ
j |) 2 +τ
(1
+
|y
−
x
(1
+
|y
−
x
j
j
MULTI-BUMP SOLUTIONS ON LATTICES
If y ∈ Bic ∩ Bi,m ∩ Ωi , we obtain that
n+2
n−2
P
P
γ(y)
≤
C
n−2 +τ
j
j
j
(1+|y−x |)
≤C
P
j
1
1
(1+|y−xj |)
n+2 +τ
2
4
(λl) n−2
k
(1 + |y − xj |)
On the other hand
X
1
j
(1 + |y −
xj |)
≥C
4 k
(λl) n−2
1
≤C
n−2
+τ
2
n+2
+τ
2
n+2 +τ + 4 (τ −k)
2
n−2
,
as τ ≥ k.
c
If y ∈ Bi,m
∩ Ωi , we have first
n+2
! n−2
X
1
j
1
1
(1+|y−xj |)
2
17
n+2
(1 + |y − xi |)
n+2
n+2
+ n−2
τ
2
1
(1 + |y −
xi |)
n+2
+τ
2
(1 + C[
m n−2 k .
m k
]) .
2
c
It is not hard to see that when y ∈ Bi,m
∩ Ωi , there is a constant C
independent of m such that for τ ≥ k,
1
which gives
X
j
1
n+2
(1 + |y − xi |)
4
τ
n−2
γ(y)
(1 + |y − xj |)
n−2
+τ
2
m n−2 k ≤ C
n+2
! n−2
≤C
(λl)
X
j
4
k
n−2
(1 + C[
m k
])
2
1
1
(1 + |y − xj |)
n+2
+τ
2
4
(λl) n−2 k
c
when y ∈ Bi,m
∩ Ωi . Therefore, we obtain that
4
kχΩ− |W̄m + φ| n−2 vk∗∗ ,
When Wm > |φ| and for n ≥ 5,
4
kWmn−2
−1
4
4
kφ n−2 vk∗∗ ≤ Ckφk∗n−2 kvk∗ .
2
2
4
|ǫWm + φ|vk∗∗ ≤ kWmn−2 φ n−2 vk∗∗ + |ǫ|kWmn−2 vk∗∗
2
≤ C(kvk∗ kφk∗n−2 + |ǫ|kvk∗).
Combining the above estimates, we deduce that
2
4
4
4
kKλ v{ (W̄m + φ)+ n−2 − Wmn−2 }k∗∗ ≤ Ckvk∗ (kφk∗n−2 + kφk∗n−2 + |ǫ|).
18
Y.Y. LI, J. WEI, AND H. XU
Similarly, we have
4
4
X
4
4
kKλ (
ηj σj ){ (W̄m + φ)+ n−2 −Wmn−2 }k∗∗ ≤ C|η|(|ǫ|+kφk∗+|ǫ| n−2 +kφk∗n−2 ).
j
Hence by Lemma A.7, we have
(2.30)
2
4
4
kṽk∗ ≤ Ckvk∗(kφk∗n−2 + |ǫ|) + C|η|(|ǫ| + kφk∗ + |ǫ| n−2 + kφk∗n−2 )
2
n−2
≤ Ck(v, η)k(kφk∗
+ |ǫ| +
kφk∗
λτ −1
4
4
+
|ǫ| n−2
λτ −1
+
kφk∗n−2
λτ −1
).
Next we estimate (2.27). It is not hard to see that from Lemma A.1,
Lemma A.2 and Lemma A.3, we can deduce that
4
4
R
| Rn Kλ σi (ηWm ){ (W̄m + φ)+ n−2 − Wmn−2 }|
4
4
≤ C|η|(|ǫ| + |ǫ| n−2 + kφk∗n−2 + kφk∗ ),
(W̄m + φ)+
4
n−2
4
Simple computations give
4
R
| Rn Kλ σi vWmn−2 |
≤
R
Rn
Ckvk∗
(1+|y−X i |)n−2
≤
Ckvk∗
λτ −1
+
Ckvk∗
≤
4k
(λl) n−2
Ckvk∗
.
λτ −1
R
4
P
j
γ(y)
(1+|y−X j |)
n−2 +τ
2
P
1
j (1+|y−X j |)n−2
4
n−2
1
1
Ωj ∩Bj (1+|y−X i |)n−2 (1+|y−X j |) n
2 +4
P R
j
4
4
− Wmn−2 = (W̄m + φ) n−2 − Wmn−2 + χΩ− |W̄m + φ| n−2 .
∪s (Ωs ∩Bsc )
1
(1+|y−X i |)n−2
P
j
1
(1+|y−X j |)
n−2 +4+τ − 4k
2
n−2
Similarly we obtain
Z
4
−1
Ckvk∗ kφk∗
|
Kλ σi vWmn−2 φ| ≤
,
λ2τ −2
Rn
and
4
n−2
Z
4
kvk
kφk
∗
∗
.
|
Kλ σi v|φ| n−2 | ≤ C τ −1
λ
λτ −1
Rn
In view of (2.29) and the above estimates, we obtain that
MULTI-BUMP SOLUTIONS ON LATTICES
|
R
Rn
≤
Kλ σi v{ (W̄m + φ)+
Ckvk∗
(|ǫ|
λτ −1
+
kφk∗
λτ −1
+
Therefore we have
λτ −1 |η̃| ≤ Ckvk∗ (|ǫ| +
(2.31)
4
4
n−2
kφk∗
λτ −1
kφk∗
λτ −1
− Wmn−2 }|
4
n−2
+
).
kφk∗
λτ −1
4
4
19
4
n−2
)
+C|η|λτ −1(|ǫ| + |ǫ| n−2 + kφk∗n−2 + kφk∗)
4
4
≤ Ck(v, η)k(|ǫ| + |ǫ| n−2 + kφk∗n−2 + kφk∗ ).
Thus by (2.30) and (2.31), we obtain
2
4
4
k(ṽ, η̃)k ≤ Ck(v, η)k(|ǫ| + |ǫ| n−2 + kφk∗n−2 + kφk∗n−2 + kφk∗ ),
which yields that
4
2
4
(2.32) kN ′ (φ, ǫ) − N ′ (0, 0)k ≤ C(|ǫ| + |ǫ| n−2 + kφk∗n−2 + kφk∗n−2 + kφk∗ ).
Step 4. Set θ =
1
2
and a =
C
λ
n+2 −τ
2
with C so large that kN ′ (0, 0)−1 N(0, 0)k ≤
(1 − θ)a. For k(φ, ǫ)k ≤ a, it follows from our estimate (2.32) that
2
kN ′ (0, 0)−1 kkN ′ (φ, ǫ) − N ′ (0, 0)k ≤ Ca n−2 < θ
for λ large. The condition of Lemma 2.1 is satisfied and the existence and
uniqueness of φ(P, Λ) and ǫ(P, Λ) follow from the lemma. The C 1 dependence follows from (2.18), (2.32)and the fact that N has C 1 dependence
on P , ǫ, Λ and φ.
3. Solving a finite dimensional problem
In this section, we will choose the positive constants C1 , C2 and integer
l0 and solve (2.4) and (2.5) for some P i ∈ B 1 (X i ) and Λi ∈ (C1 , C2 ) when
2
l ≥ l0 .
To this end, we need some preliminary computations.
Lemma 3.1.
(3.1)
(1 + ǫi )
−1
∂J
=−
∂P i j
Z
Rn
n+2
Kλ σin−2 Zi,j + o(
1
).
λβ
20
Y.Y. LI, J. WEI, AND H. XU
Proof. The left hand side of (3.1) equals
P
∂σi
∂J
(1 + ǫi )−1 ∂P
i =
s6=i (1 + ǫs )h ∂P i j , σs i
j
R
−
n+2
Rn
P
=
R
−
Kλ ((Wm + ǫWm + φ)+ ) n−2 Zi,j
s6=i (1
∂σi
+ ǫs )h ∂P
i , σs i −
j
n+2
R
Rn
Kλ (Wm + ǫWm + φ) n−2 Zi,j
n+2
Ω−
Kλ |Wm + ǫWm + φ| n−2 Zi,j .
BY Proposition 2.1, |ǫ| is small. Therefore in Ω− , |φ| ≥ W2m . From
Lemma A.5 and the estimates in Lemma A.11, we deduce that
R
R
n+2
n+2
| Ω− Kλ |Wm + ǫWm + φ| n−2 Zi,j | ≤ C |φ|≥ Wm |φ| n−2 |Zi,j |
2
n+2
≤ Ckφk∗n−2
1
≤
n +τ n+2
n−2
(λl) 2
C
2(n+2)
n+ n−2
λ
.
Lemma 3.1 now follows from Lemma A.11 and Proposition 2.1.
Lemma 3.2.
P
∂σi
∂J
= j6=i h ∂Λ
, σj i
(1 + ǫi )−1 ∂Λ
i
i
(3.2)
−
R
Rn
P
Kλ
n+2
n−2
k
σk
4
n+2 n−2
σ
n−2 i
+
P
j6=i
σj Zi,n+1 + o( λ1β ).
Proof. As before the left hand side equals
P
∂σi
∂J
(1 + ǫi )−1 ∂Λ
=
j6=i (1 + ǫj )h ∂Λi , σj i
i
−
=
−
where
R
R
Rn
P
Kλ (W̄m + φ|)+
j6=i (1
R
Ω−
n+2
n−2
∂σi
, σj i −
+ ǫj )h ∂Λ
i
n+2
Zi,n+1
R
Rn
4
Kλ |W̄m + φ| n−2 (W̄m + φ)Zi,n+1
Kλ |W̄m + φ| n−2 Zi,n+1 ,
4
Kλ |W̄m + φ| n−2 (W̄m + φ)Zi,n+1 =
Rn
+ n+2
n−2
R
+O(1)
4
Rn
R
Kλ Wmn−2 (ǫWm + φ)Zi,n+1
n+2
|φ| n−2 |Zi,n+1| + O(1)
|φ|≥Wm
R
n+2
Kλ Wmn−2 Zi,n+1
Rn
6−n
Wmn−2 |ǫWm + φ|2 |Zi,n+1|.
|φ|≤Wm
R
MULTI-BUMP SOLUTIONS ON LATTICES
21
Here O(1) means a number uniformly bounded independent of m or l.
C
Recall that |Zi,n+1 | ≤ (1+|y−P
i |)n−2 . When λ is large, by Lemma A.5,
if |φ| ≥ Wm , then y ∈ (∪j (Bj ∩ Ωj ))c := Ω. By Lemma A.1 and similar
argument as in Lemma A.11, we deduce that
|
Z
Kλ |W̄m + φ|
Ω−
Z
6−n
n−2
Wm
|φ|≤Wm
n+2
n−2
Z
Zi,n+1 | ≤ C
C
n+2
Ω
|φ| n−2 |Zi,n+1| ≤
λn−1+
2(n+2)
n−2
.
kφk2∗
C
|ǫWm + φ| |Zi,n+1| ≤ C( 2τ −2 + |ǫ|2 ) ≤ n .
λ
λ
2
Since hφ, Zi,n+1i = 0, by Lemma A.6,
Z
Z
4
4
1
C
n−2
Kλ Wm φZi,n+1 =
(Kλ − 1)σin−2 φZi,n+1 + o( n ) ≤ n .
λ
λ
Rn
Rn
Similarly
4
R
Kλ Wmn−2 ǫWm Zi,n+1
Rn
=
R
n+2
Rn
(Kλ − 1)ǫi σin−2 Zi,n+1 +
C|ǫ|
(λl)n−2
≤ o( λ1n ).
In the region y ∈ Ωj ∩ Bjc
C
n+2
Wmn−2 (y) ≤
(λl)
4
k
n−2
X
s
1
4
(1 + |y − X s |)n+2− n−2 k
Therefore by Lemma A.1 we have that for all n ≥ 5
n+2
R
| ∪j (Ωj ∩Bc ) Kλ (y)Wmn−2 Zi,n+1 dy| ≤
.
j
C
4 k
(λl) n−2
≤
≤
C
R
∪j (Ωj ∩Bjc )
4 k
(λl) n−2
C
(λl)n
P
s6=i
P
1
(1+|y−X s |)
1
|X i −X s |
n−2
2
4
+
C
(λl)n
4
n+2
|Wmn−2 − σjn−2 −
2
≤ C Ŵm,j
σjn−2
1
n+2 − 4 k
n−2
(λl) 2
For y ∈ Ωj ∩ Bj , we have
n+2
1
dy
(1+|y−X i |)n−2
4 k
n+2− n−2
s
−1
n+2 n−2
σ Ŵm,j |
n−2 j
n
n
n−2
≤ Ŵm,j
σjn−2
−1
.
22
Y.Y. LI, J. WEI, AND H. XU
We can show for j 6= i,
n+2
n+2
R
R
n−2
n−2
K
(y)W
Z
dy
=
Zi,n+1 dy
K
(y)σ
m
λ
i,n+1
λ
j
Ωj ∩Bj
Ωj ∩Bj
+ (λl)2 |X iC−X j |n−2 =
R
n+2
Kλ (y)σjn−2 Zi,n+1dy +
Rn
When j = i,
n+2
R
n−2
K
(y)W
Zi,n+1 dy
m
λ
Ωi ∩Bi
n+2
= Ωi ∩Bi Kλ (y) σin−2 +
R
n+2
= Rn Kλ (y) σin−2 +
R
4
n+2 n−2
σ
n−2 i
X
j6=i
ǫj h
j6=i
4
n+2 n−2
σ
n−2 i
Together with
|
P
P
j6=i
C
.
(λl)2 |X i −X j |n−3
σj Zi,j dy +
σj Zi,n+1 dy +
C
(λl)n−1
C
.
(λl)n−1
∂σi
C|ǫ|
1
, σj i ≤
|
=
o(
),
∂Λi
(λl)n−2
λβ
we can easily deduce the estimate (3.2).
By the estimates in [5],
Z
n+2
n
∂ǫij
1
∂σi
= C4
+ O(ǫijn−2 log ǫ−1
σjn−2
ij )
∂λi
∂λi
λi
Rn
n R
1
where C4 = (n(n − 2)) 2 Rn
n+2 dy and
2
(1+|y| )
ǫij =
2
λi λj
+
+ λi λj |P i − P j |2
λj
λi
− n−2
2
,
for i 6= j.
Using Lemma 3.1 and Lemma A.9, we infer that (2.4) is equivalent to
(3.3)
Dn,β aj
1
Λiβ−2 λβ
(P i j − X i j ) = O(
|P i − X i |2
1
) + o( β ),
β
λ
λ
for all i = 1, ..., (m + 1)k and j = 1, ..., n.
By Lemma 3.2, Lemma A.10, Lemma A.12 and Lemma A.13, we can
derive that (2.5) is equivalent to
(3.4)
X (n − 2)C4 Aij Λj
C3
1
|P i − X i |β−1
−
=
o(
)
+
O(
).
n
β+1 β
β
β
2 (λl)n−2
λ
λ
2(Λ
Λ
)
Λ
λ
i
j
i
j6=i
MULTI-BUMP SOLUTIONS ON LATTICES
23
In the above, A := {Aij } is a (m + 1)k × (m + 1)k matrix associated to
the lattice Xl,m (or X1,m ), given as follows


 0 if i = j
A = (Aij ) =
n−2

λl

if i 6= j.
i
j
|X −X |
− n−2
2
If we take bi = Λi
X
(3.5)
Aij bj −
j6=i
, then we see that (2.5) is equivalent to
2β
|P i − X i |β−1
2C3
−1
+ o(1) + O(
) bin−2 = 0.
(n − 2)C4
λ
(m+1)k
→ R defined by
Now we consider the functional F : R+
2β
C3 X n−2
1
bi , for b = (b1 , ..., b(m+1)k ).
(3.6)
F (b) = bt Ab −
2
βC4 i
Since C3 > 0 and β > n − 2, the maximum of F will give a solution to
the system
2β
X
2C3
−1
Fi (b) =
Aij bj −
bin−2 = 0, i = 1, ..., (m + 1)k .
(n − 2)C4
j6=i
Let B̄m = (b̄1 , ..., b̄(m+1)k ) be a solution to the above system.
Lemma 3.3. There exist positive constants C5 < C6 independent of m,
such that C5 ≤ |b̄i | ≤ C6 for all i = 1, ..., (m + 1)k .
Proof. For each Fi , for any integer m ≥ 1, without loss of generality, we
can assume that b̄1 ≤ b̄i ≤ b̄2 for all i = 1, ..., (m + 1)k . From the equation
F2 (B̄m ) = 0, summing in j, we can get
2β
b̄2n−2
where C =
(n−2)C4
2C3
P
j6=i
−1
≤ C max b̄j ≤ C b̄2 ,
Aij and
j6=i
P
j6=i
Aij can be controlled by
dy
.
Rk 1+|y|n−2
R
Similarly from the equation F1 (B̄m ) = 0, summing in j, we deduce that
2β
(n − 2)C4
(n − 2)C4 X
−1
Aij b̄1 ≥
b̄1 .
b̄1n−2 ≥
2C3
2C3
j6=i
From the above two inequalities, we conclude that C5 ≤ b̄i ≤ C6 for all
i = 1, ..., (m + 1)k .
24
Y.Y. LI, J. WEI, AND H. XU
2β
By the form of F and using the fact that 2 < n−2
< 4, we see that the
2
Hessian matrix at B̄m D F (B̄m ) is negative definite (this ensures that B̄m
is unique). We will show that the inverse matrix of D 2 F (B̄m ) is uniformly
bounded independent of m when l is large enough.
k
Take X = (x1 , ..., x(m+1)k ) in R(m+1) ,
2β
X
2β
2C3
−2
Aij xj − (
D 2 (F (B̄m ))X i =
− 1)
b̄in−2 xi .
n−2
(n − 2)C4
j6=i
Consider the i with largest | xb̄ii | (from Lemma 3.3, |xi | ≥ C|X|). By the
equation Fi (B̄m ) = 0, as 2β > 2(n − 2), we obtain
2β
2β
2C3
|( n−2
− 1) (n−2)C
b̄ n−2
4 i
≥|
P
j6=i
−2
Aij b̄j | × | xb̄ii | ≥
This implies that
2β
2C3
b̄ n−2
xi | ≥ | (n−2)C
4 i
P
j6=i
Aij |xj | ≥ |
−1 x
i
b̄i
P
|
j6=i
Aij xj |.
2β
P
−2
2β
n−2
2C3
x
|
−
|
− 1) (n−2)C
b̄
| D 2 (F (B̄m ))X i | ≥ |( n−2
i
i
j6=i Aij xj |
4
2β
2β
2C3
− 1) (n−2)C
b̄ n−2
≥ | xb̄ii |( n−2
4 i
2β
2β
2C3
− 2) (n−2)C
b̄ n−2
= ( n−2
4 i
−2
b̄i − | xb̄ii |(
−1 x
| b̄ii |
P
j6=i
Aij b̄j ),
(by Fi (B̄m ) = 0)
≥ C|X| (by Lemma 3.3),
where C only depends on C3 , C4 , C5 , C6 . Hence we get
|D 2 F (B̄m )X| ≥ C|X|.
Similarly, we can also show that
|D 2 F (B̄m )X| ≤ C|X|.
Thus we obtain that | D 2 F (B̄m )
with maximum norm | · |.
−1
k
X| ≤ C|X| for all X ∈ R(m+1)
Proof of Theorem 1. By (3.3), (2.4) is equivalent to
(3.7)
P i − X i = O(|P i − X i |2 ) + o(1),
From (3.5) if we let t = b − B̄m ∈ R
(3.8)
(m+1)k
for all i,
, then (2.5) is equivalent to
D 2 F (B̄m )t = O(|t|2) + o(1) + O(max |P i − X i |2 ).
MULTI-BUMP SOLUTIONS ON LATTICES
25
k
For R(m+1) ×(n+1) , equipped with maximum norm, we can choose a C >
0 large but independent of m and l. When l is large enough, (3.7) and
(3.8) define a continuous map from
k
B := BCo(1) (X 1 ) × ... × BCo(1) (X (m+1) ) × BCo(1) (B̄m ) → B.
By Brouwer fixed point theorem, we can solve equations (3.7) and (3.8)
k
near (X 1 , ..., X (m+1) , Bm ) with
|P i − X i | = o(1),
Therefore we have solved
(3.9)
−
∂J
∂Λi
= 0,
|b − Bm | = o(1).
∂J
∂P i j
2
|Λ − Bm n−2 | = o(1),
when λ large enough.
= 0 with
|P i − X i | = o(1),
By Lemma 3.3, we can now choose positive constants C1 < C2 which
only depend on C5 and C6 and are independent of m and l. Then we can
take integer l0 large enough such that the (P, Λ) given in (3.9) satisfies
P i ∈ B 1 (X i ) and Λi ∈ (C1 , C2 ) for all i and l ≥ l0 . Therefore a solution
2
to equation (1.4) is guaranteed.
Now we are ready to prove Theorem 2.
Proof of Theorem 2. Let {um } denote the solutions of (1.1 given by
Theorem 1 with l ≥ l0 large and fixed. For each m, we can find xm ∈ Xl,m ,
such that
i
∪∞
m=1 (Xl,m − xm ) = Xl .
Let
(Sλ ûm )(x) = (Sλ um )(x + xm ).
Then ûm satisfies the same equation as um due to the periodicity of K.
We will show that there exists some constant C(l), independent of m, such
that
(3.10)
ûm (x) ≤ C(l),
∀ x ∈ Rn .
Once we have (3.10), we then deduce, by elliptic estimates, that for any
R > 1, there exists some constant C2 (l), independent of m, such that
(3.11)
kûm kC 3 (BR ) ≤ C2 (l),
∀ m = 1, 2, 3, ...
This implies that we can pass to a subsequence {ûmi } such that
2
ûmi → u in Cloc
(Rn )
for some non-negative function u ∈ C 2 (Rn ). Clearly u satisfies
n+2
−∆u = K(x)u+n−2 ,
in Rn .
26
Y.Y. LI, J. WEI, AND H. XU
By the form of u given by Theorem 1, u can not be identically zero,
provided that l is large (but independent of m). In fact from the form
of u given by Theorem 1, u is clearly bounded from below in B1 (0) by
a positive constant independent of m. Therefore, by strong maximum
principle, u > 0 in Rn .
It remains to prove (3.10). This follows from the form of Sλ um (given
by Theorem 1). In fact by the estimates on kφk∗ , when k < n−2
,
2
X
1
≤ Ckφk∗.
kφkL∞ (Rn ) ≤ kφk∗
n−2
i |) 2 +τ
(1
+
|x
−
X
i
where C doesn’t depend on m. For the same reason, we also have
X
k
σP i ,Λi kL∞ (Rn ) ≤ C,
X i ∈Xl,m
where C doesn’t depend on m.
Thus it follows from the form of solution given by Theorem 1, that
kSλ um kL∞ (Rn ) ≤ C.
By the form of Sλ , we get kum kL∞ (Rn ) ≤ Cλ
thus established.
n−2
2
(n−2)2
= Cl 2(β−n+2) , (3.10) is
4. Proof of Theorem 3
We first give a lemma which is used in the proof of Theorem 3.
α
Lemma 4.1. For n ≥ 3, 0 < α < 1, let f ∈ Cloc
(Rn ) be nonnegative
n
2
n
outside a compact set of R . Assume that u ∈ C (R ) satisfies
and
−∆u = f
in Rn ,
lim inf u(x) > −∞.
|x|→∞
Then, for some constant a ≥ min(0, lim inf |x|→∞ u(x)),
Z
f (x̃)dx̃
1
u(x) =
+ a, ∀x ∈ Rn ,
n(n − 2)ωn Rn |x − x̃|n−2
where ωn is the volume of a unit ball in Rn .
Proof. By adding − min(0, lim inf |x|→∞ u(x)) to u, we may assume, without loss of generality, that lim inf |x|→∞ u(x) ≥ 0.
Let
Z
1
f (x̃)dx̃
ui (x) =
, i = 1, 2, 3, ....
n(n − 2)ωn Bi |x − x̃|n−2
MULTI-BUMP SOLUTIONS ON LATTICES
27
We know that
∆(u − ui ) = 0 in Bi ,
(4.1)
and, using the fact that f is nonnegative outside a compact set,
Clearly
ui ≤ ui+1,
∆(u − ui ) ≤ 0,
in Rn for large i.
lim inf (u − ui )(x) ≥ 0.
|x|→∞
Thus, by the maximum principle,
u − ui ≥ 0 in Rn
for large i.
Using the Fatou’s Lemma, we obtain
Z
f (x̃)dx̃
1
≤ lim ui (x) ≤ u(x).
n(n − 2)ωn Rn |x − x̃|n−2 i→∞
Now, by the Lebesgue dominated convergence theorem,
Z
f (x̃)dx̃
1
≤ u(x), ∀x ∈ Rn .
lim ui (x) =
n−2
i→∞
n(n − 2)ωn Rn |x − x̃|
For every R > 0,
∆(u − ui ) = 0 in B2R ,
∀i > 2R.
We know that {u − ui }, for large i, is a non-increasing sequence of nonnegative harmonic functions in B2R . In particular, {u − ui } is uniformly
bounded in B2R . By the interior derivative estimates of harmonic functions, the convergence of {u − ui } is C 2 in B2R . Thus {u − ui } converges
2
to some function ξ in Cloc
(Rn ). The entire nonnegative harmonic function
ξ is a constant, denoted by a. Lemma 4.1 is established.
Proof of Theorem 3. We prove it by contradiction. Let u be a C 2 solution of (1.1) satisfying (1.10) for some R, ǫ > 0 and 0 ≤ i ≤ k. We divide
the proof into three steps.
Step 1. For any a > 0, we have
sup{u(x)|x ∈ Rn ,
dist(x, Rki ) < a} < ∞.
Suppose not, by making a translation according to the periods of K, we
may assume that there exists |xj | ≤ a + 1, such that uj , the corresponding
translations of u, satisfies

n+2

 −∆uj = Kujn−2 , uj > 0 in Rn ,

 u (x ) → ∞
j
j
28
Y.Y. LI, J. WEI, AND H. XU
and
inf sup uj ≥ ǫ.
(4.2)
x∈Rki BR (x)
By Theorem 1.2 in [15], there exists a positive constant C(K, a) such
that
Z
2n
|∇uj |2 + ujn−2 ≤ C(K, a).
B2a (0)
By Proposition B.1, {uj }, after passing to a subsequence, only has isolated simple blow up points in Rn . Let S be the set of blow up points in
Rn . We know that S 6= ∅. Proposition 4.2 of [34], applied to translations
of {uj }, shows that there exists a δ > 0, such that
inf
x,y∈S,x6=y
|x − y| ≥ δ.
Passing to a subsequence and replacing xj by some nearby points if
necessary, we may assume that xj → x̄ ∈ S is an isolated simple blow up
point. Thus by Proposition 2.3 in [34], that
2
uj (xj )uj → h Cloc
(Rn \ S),
where h is a positive harmonic function on Rn \ S and has a singularity
at each point in S. By the proof of Theorem 4.2 in [34], S can not have
more than one point, so S = {x̄}, and uj → 0 uniformly on any compact
subset of Rn \ {x̄}. This contradicts (4.2).
Step 2. For any a > 0,
By step 1,
inf{u(x)|x ∈ Rn ,
sup
dist(x,Rki )<2a
u(x) = C(a) < ∞.
Since
−∆u = Ku
4
4
dist(x, Rki ) < a} > 0.
n+2
n−2
= Ku
4
n−2
u,
and |Ku n−2 | ≤ (sup K)C(a) n−2 if dist(x, Rki ) < 2a, We apply the Harnack
inequality to obtain
sup u ≤ C(a, sup K) inf u,
Ba (x)
Ba (x)
∀x ∈ Rki .
We may assume without loss of generality that a ≥ R. Then, in view
of (1.10), we have
sup u ≥ ǫ, ∀x ∈ Rki .
Ba (x)
Step 2 is established.
MULTI-BUMP SOLUTIONS ON LATTICES
29
Taking a = 1 in step 2, we have, for some positive constant b,
(4.3)
∀x such that dist(x, Rki ) < 1.
u(x) ≥ b,
Step 3. When k ≥
n−2
,
2
(1.1) has no C 2 solution satisfying (4.3).
By Lemma 4.1,
(4.4)
1
u(x) =
n(n − 2)ωn
Z
n+2
Rn
K(x̃)u n−2 (x̃)
dx̃ + a,
|x − x̃|n−2
∀x ∈ Rn ,
where a ≥ 0. We will show that a = 0.
Since u > 0 in Rn and inf K > 0, (4.4) implies
n+2 Z
dx̃
(inf K)a n−2
,
u(x) ≥
n(n − 2)ωn Rn |x − x̃|n−2
R
dx̃
therefore a = 0, since Rn |x−x̃|
n−2 = ∞.
From (4.4) with a = 0 and (4.3), we have, for some constant C > 1,
Z
dx̃
1
, ∀x ∈ Rn .
(4.5)
u(x) ≥
C dist(x̃,Rki )<1 |x − x̃|n−2
If k ≥ n − 2, the right hand side of the above is ∞, which is impossible.
Now we treat the remaining case: n−2
≤ k < n − 2. For convenience,
2
we write Rn = Rk × Rn−k . For any x ∈ Rn , x = (y, z) ∈ Rk × Rn−k and
u(x) = u(y, z). We show that, for some constant C > 1,
1
, ∀(y, z) ∈ Rki × Rn−k .
(4.6)
u(y, z) ≥
C(1 + |z|)n−2−k
By (4.5),
dỹdz̃
,
|z̃|≤1 |(y,z)−(ỹ,z̃)|n−2
u(y, z) ≥
1
C
≥
1
C
≥
1
2i C
≥
ωn−k
2i C(|z|+1)n−2−k
R
Rki
R
dξ
Rki (1+|ξ|2 ) n−2
2
R
dξ
Rk (1+|ξ|2 ) n−2
2
R
dz̃
,
|z̃|≤1 |z−z̃|n−k−2
R
dz̃
,
|z̃|≤1 |z−z̃|n−k−2
R
dξ
Rk (1+|ξ|2 ) n−2
2
R
≥
1
.
C(1+|z|)n−2−k
ỹ−y
Here we have made a change of variables ξ = |z̃−z|
and have used the
k
fact that for every fixed z̃ − z 6= 0 and y ∈ Ri , the set
{
ỹ − y
|ỹ ∈ Rki } ⊃ Rki .
|z̃ − z|
30
Y.Y. LI, J. WEI, AND H. XU
Let v be the spherical average of u defined by
Z
1
udS,
v(x) = v(|x|) = v(r) =
|∂Br (0)| ∂Br (0)
then by Jensen’s inequality,
n+2
in Rn .
−∆v ≥ (inf K)v n−2 ,
By some elementary argument, see e. g. [14], we have, for some constant
C > 0,
v(x) ≤
(4.7)
For k >
n−2
,
2
C
(1 + |x|)
n−2
2
for any x ∈ Rn .
,
we obtain (r = |x|), using (4.6) and (4.7),
C
n−2
(1+r) 2
≥ v(r) =
Rr
≥
1
Cr n−1
≥
2i Cr n−1
≥
1
C
≥
Cr n−2−k
≥
1
,
C(1+r)n−2−k
1
0
R
da
1
|∂Br (0)|
R
∂Br (0)
udS
√
{|y|=a}∩Rki ,|z|= r 2 −a2
Rr
1
ak−1 (
0 (1+ r 2 −a2 )n−2−k
√
u(y, z)
√
r 2 − a2 )n−k−1da
√
sk−1 ( 1−s2 )n−k−1
√
ds
0 (1+r 1−s2 )n−2−k
R1
1
√
sk−1 ( 1−s2 )n−k−1
√
ds,
0 (1+ 1−s2 )n−2−k
R1
for r ≥ 1
for r ≥ 1.
Sending r to ∞ leads to a contradiction.
For k = n−2
and 0 ≤ i ≤ k = n−2
, we derive from (4.4) (notice that
2
2
a = 0) and (4.6) that, for any (y, z) ∈ Rki × Rn−k ,
R
dỹdz̃
u(y, z) ≥ C1 Rk ×Rn−k
(n−2−k) n+2
|(y,z)−(ỹ,z̃)|n−2 (1+|z̃|)
i
(4.8)
dξ
≥
1
C
≥
max(1,log |z|)
R
Rki
n−2
(1+|ξ|2 ) 2
C(1+|z|)
n−2
2
R
Rn−k
n−2
dz̃
n+2
(n−2−k) n−2
|z̃−z|n−k−2 (1+|z̃|)
by Lemma A.2.
MULTI-BUMP SOLUTIONS ON LATTICES
31
and 0 ≤ i ≤ k,
By (4.7) and (4.8), we obtain when k = n−2
2
R
1
1
udS
n−2 ≥ v(r) = |∂B (0)|
∂Br (0)
r
(1+r)
2
Rr
≥
1
Cr n−1
≥
2i Cr n−1
≥
1
C
≥
1
C
≥
1
0
da
R
√
{|y|=a}∩Rki ,|z|= r 2 −a2
√
√
max(1,log r 1−a2 ) n−2 −1
2
(
a
r2
√
n−2
0 (1+ r 2 −a2 ) 2
Rr
√
√
max(1,log r 1−s2 ) n−2 −1
s 2 ( 1
√
0 (1+r 1−s2 ) n−2
2
R1
√
log 23r
√
0 (1+r 1−s2 ) n−2
2
R1
2
r
2
n−2
(1+r) 2
log
u(y, z)
s
n−2
−1
2
n
− a2 ) 4 da
n
− s2 ) 4 ds
√
n
( 1 − s2 ) 4 ds,
for r ≥ 10
for r ≥ 10.
,
We also arrive at a contradiction when r → ∞. Thus Theorem 3 is
proved.
,
From the proof of Theorem 3, it is easy to see that when k < n−2
2
0 ≤ i ≤ k and K has a positive lower bound, (1.1) does not admit a
solution u satisfying
lim (1 + |z|)
|z|→∞
In some sense,
of (1.1).
n−2
2
n−2
2
u(y, z) = ∞,
uniformly in y ∈ Rki .
is a threshold value for the decay power of solutions
, suppose that K ≥ 0, but not identically
Lemma 4.2. Let 1 ≤ k < n−2
2
equal zero and K is bounded from above. Let u be a positive solution of
(1.1). Assume, for some constants τ > 0, that
(4.9)
sup (1 + |z|)
(y,z)∈Rn
n−2
+τ
2
u(y, z) < ∞.
Then
(4.10)
sup (1 + |z|)n−2−k u(y, z) < ∞.
(y,z)∈Rn
Proof. When τ ≥ n−2
− k, (4.10) is obvious. Now we consider the case
2
−
k.
By
(4.4)
(notice that a = 0) and (4.9), we obtain that
0 < τ < n−2
2
32
Y.Y. LI, J. WEI, AND H. XU
for some constant C > 0,
R R
u(y, z) ≤ C Rk Rn−k
≤C
Therefore if
A.2, we have
n−2
2
+
1
|(y,z)−(ỹ,z̃)|n−2
1
Rn−k |z−z̃|n−k−2
R
n+2
τ
n−2
1
(1+|z̃|)
1
n−2 + n+2 τ +2
2
n−2
n−2 + n+2 τ +2
n−2
(1+|z̃|) 2
dỹdz̃,
dz̃.
6= n − k − 2, applying the first part of Lemma
C
u(y, z) ≤
.
(1 + |z|)
n+2
+ n−2
τ > n − k − 2, we are done. Therefore we only need to
If n−2
2
consider the case n−2
+ n+2
τ < n − k − 2 if they are not equal. In this
2
n−2
case, we get
C
u(y, z) ≤
n−2
n+2 .
(1 + |z|) 2 + n−2 τ
n+2
τ and τi = n+2
τ . Obviously, {τi } is an increasing seLet τ1 = n−2
n−2 i−1
quence and hence we can iterate till we get n−2
+ n+2
τ ≥ n − k − 2. If
2
n−2 i
n−2
n+2
n−2
n+2
+ n−2 τi > n − k − 2, then we are done. If 2 + n−2 τi = n − k − 2, i.
2
n−2
− k) n+2
, we can apply the second part of Lemma A.2 to get
e., τi = ( n−2
2
(4.11)
u(y, z) ≤ C
n+2
min(n−k−2, n−2
+ n−2
τ)
2
max(1, log |z|)
,
(1 + |z|)n−k−2
∀(y, z) ∈ Rn .
− k), then n−2
+ n+2
τ
≥ n − k − 2. By
Choose any τi+1 ∈ (τi , n−2
2
2
n−2 i+1
(4.11), we have
C
.
u(y, z) ≤
n−2
(1 + |z|) 2 +τi+1
Since n−2
+ n+2
τ
> n − k − 2, by one more iteration, we get the
2
n−2 i+1
conclusion.
Remark 4.1. We easily see from the proof of Theorem 1 and Lemma 4.2
that solutions constructed in Theorem 1 and Theorem 2 satisfy (4.10).
In the following, for every k ∈ [1, n−2
), we give examples of positive
2
smooth function K such that there is a solution u of (1.1) satisfying (1.10)
for some R, ǫ > 0 and all i ∈ [0, k].
Let (y, z) ∈ Rk × Rn−k and u(y, z) = v(z) =
1
(1+|z|2 )
n−2
4
. Direct calcula-
tion shows that
n−2
−∆u(y, z) = −∆z v(z) =
2
n−2
n−2
−k+
2
2(1 + |z|2 )
n+2
u(y, z) n−2 .
MULTI-BUMP SOLUTIONS ON LATTICES
Moreover u decays like
1
(1+|z|)
n−2 . Here K(y, z) =
2
which is periodic in y.
n−2
2
33
n−2
2
−k+
n−2
2(1+|z|2 )
5. Appendix A
In this section, we present the proof of some technical lemmas.
For xi xj , y ∈ Rn , define
gij (y) =
1
(1 + |y − xi
|)α (1
+ |y − xj |)β
where xi 6= xj and α > 0 and β > 0 are two constants.
We first prove a lemma which slightly improves the Lemma B.1 in [49].
Lemma A.1. For any constant τ ∈ [0, min(α, β)], we have
1
1
2τ
+
.
gij (y) ≤
(1 + |xi − xj |)τ (1 + |y − xi |)α+β−τ
(1 + |y − xj |)α+β−τ
Proof. Let d = |xi − xj |. If y ∈ B d (xi ), then
2
d
|y − xj | ≥ ,
2
|y − xj | ≥ |y − xi |,
which implies
gij (y) ≤
1
1
,
1 τ
(1 + 2 d) (1 + |y − xi |)α+β−τ
y ∈ B 1 d (xi ).
2
Similarly, we have
1
1
, y ∈ B 1 d (xj ).
1
τ
2
(1 + 2 d) (1 + |y − xj |)α+β−τ
Now we consider y ∈ Rn \ B 1 d (xi ) ∪ B 1 d (xj ) . Then we have |y −xi | ≥
2
2
d, |y − xj | ≥ d. We may also assume that |y − xi | ≥ |y − xj |. This yields
that
1
1
gij (y) ≤
.
τ
(1 + d) (1 + |y − xj |)α+β−τ
The result of the Lemma follows easily from above inequalities.
gij (y) ≤
Lemma A.2. [49] For any constant 0 < τ with τ 6= n − 2, there exists a
constant C = C(n, τ ) > 1 such that
Z
1
C
1
≤
dz ≤
.
min(τ,n−2)
n−2
2+τ
C(1 + |y|)
(1 + |z|)
(1 + |y|)min(τ,n−2)
Rn |y − z|
34
Y.Y. LI, J. WEI, AND H. XU
When τ = n − 2, there exists a constant C = C(n) > 1 such that
max(1, log |y|)
≤
C(1 + |y|)n−2
Z
1
Rn
|y −
z|n−2 (1
+
|z|)n
dz ≤
C max(1, log |y|)
.
(1 + |y|)n−2
Proof. This follows from a simple modification of the proof of Lemma B.2
in [49]. So we omit the details.
(m+1)k
Recall that for Xl,m = {X i }i=1 , Ωi = {y ∈ Rn , such that |y − X i | ≤
|y − X j |, for all j 6= i}, Bi = Bλl (X i ) and Bi,m = Bmax( m4 ,1)λl (X i ).
The following lemma provides basic estimates and will be used frequently in the sequel.
Lemma A.3. For any θ > k, there exists a constant C(θ, k, n) > 1 independent of m, such that if y ∈ Bi ∩ Ωi ,
(A1)
X
1
C
1
≤
≤
;
i
θ
j
θ
(1 + |y − X |)
(1 + |y − X |)
(1 + |y − X i |)θ
j
If y ∈ Bic ∩ Bi,m ∩ Ωi ,
(A2)
X
1
1
C
≤
≤
;
i
θ−k
k
j
θ
i |)θ−k (λl)k
C(1 + |y − X |) (λl)
(1
+
|y
−
X
|)
(1
+
|y
−
X
j
c
and if y ∈ Bi,m
∩ Ωi ,
(A3)
X
1
Cmk
C
mk
≤
≤
≤
.
i
θ
j
θ
i
θ
C(1 + |y − X |)
(1 + |y − X |)
(1 + |y − X |)
(1 + |y − X i |)θ−k (λl)k
j
Proof. For any y ∈ Ωi , since y is closest to X i , by triangle inequality, we
have
3|y − X j | ≥ |y − X i | + |X i − X j | ≥ |y − X j |.
MULTI-BUMP SOLUTIONS ON LATTICES
35
Hence
1
j (1+|y−X j |)θ
P
≤
(A4)
≤
C
(1+|y−X i |)θ
C
(1+|y−X i |)θ
≤
≤
1+
1+
C
(1+|y−X i |)θ
C
(1+|y−X i |)θ
j 1+
|X i −X j |
(1+|y−X i |)
1
R
[−m−1,m+1]k
(1+|y−X i |)k
(λl)k
1+
1
P
R
1+
|z|≤
(1+|y−X i |)k
(λl)k
λl
|z|
1+|y−X i |
(m+1)λl
(1+|y−X i |)
,
θ
θ dz
!
1
dz
(1+|z|)θ
if θ > k.
If y ∈ Bi ∩ Ωi , the inequalities in (A1) can be easily obtained from the
above.
If y ∈ Bic ∩ Ωi , we have
(A5)
P
1
j (1+|y−X j |)θ
≥
C
(1+|y−X i |)θ
≥
C
(1+|y−X i |)θ
≥
C
(1+|y−X i |)θ
1+2
1+
−k
R
P
1
j 1+
|X i −X j |
(1+|y−X i |)
[0,[ m
]+1]k \[0,1]k (1+
2
(1+|y−X i |)k
(λl)k
R
θ
1
λl
|z|)θ
(1+|y−X i |)
([ m ]+1)λl
λl
≤|z|≤ 2
(1+|y−X i |)
(1+|y−X i |)
dz
1
dz
(1+|z|)θ
,
where the constant 2−k is due to the reason that for any point X j ∈ Xl,m ,
the integral region always contains a quadrant of [0, [ m2 ] + 1]k \ [0, 1]k if it
is not empty.
Now when y ∈ Bic ∩ Bi,m ∩ Ωi , we may assume that m ≥ 8, since
[ m ]+1
1 < [ m4 ] ≤ 2 2 , we have
Z
( m ]+1)λl
λl
≤|z|≤ 2
(1+|y−X i |)
(1+|y−X i |)
1
dz ≥
(1 + |z|)θ
1
dz > 0.
θ
1≤|z|≤2 (1 + |z|)
Z
The inequalities in (A2) follows easily from above observation and (A4).
4
λl
c
When y ∈ Bi,m
∩ Ωi , since 1+|y−X
i | ≤ m , we have
Z
Z
1
1
dz ≥
dz ≥ Cmk .
4
λl
θ
θ
m
m
[0,[ 2 ]+1]k \[0,1]k (1 + (1+|y−X i |) |z|)
[0,[ 2 ]+1]k \[0,1]k (1 + m |z|)
Then the inequalities in (A3) follows from (A4) and (A5).
36
Y.Y. LI, J. WEI, AND H. XU
and 0 < C1 < C2 <
Lemma A.4. Suppose that n ≥ 5, 1 ≤ k < n−2
2
], such that
∞. We can find a positive constant τ0 = τ0 (n, k) ∈ (k, n−2
2
for any k ≤ τ < τ0 , there exist constants θ = θ(τ, n, k) > 0 and C =
C(C1 , C2 , k, n), such that
4
R
P
n−2
1
1
W
γ(z) j
m
n−2 +τ dz ≤
Rn |y−z|n−2
j
(1+|z−X |)
Cγ(y)
P
j
1
n−2
(1+|y−X j |) 2 +τ +θ
+
C
4k
(λl) n−2
2
γ(y)
1
P
j
(1+|y−X j |)
n−2 +τ
2
.
Proof. Since n ≥ 5 and k < n−2
, using Lemma A.3, we obtain that for
2
z ∈ Bi ∩ Ωi ,
4 X
1
1
≤
C
Wmn−2
n−2
n+2
+τ
j
2
(1 + |z − X i |) 2 +2+τ
j (1 + |z − X |)
and for z ∈ Bic ∩ Ωi ∩ Bi,m
X
4 X
1
1
1
Wmn−2
≤
C
.
4
n−2
n−2
4
k
j |) 2 +τ
j |) 2 +4+τ − n−2 k
n−2
(1
+
|z
−
X
(λl)
(1
+
|z
−
X
j
j
c
For z ∈ Bi,m
∩ Ωi , we also have
4
n−2
Wm
X
j
≤C
4
1
(1 + |z − X j |)
X
1
n−2
+τ
2
≤C
mk+ n−2 k
(1 + |z − X i |)
n−2
+τ +4
2
1
4
k
n−2
4
n−2
+4+τ − n−2 k
j
2
(λl)
j (1 + |z − X |)
Now we compute
4
R
P
n−2
1
1
W
γ(z)
m
n−2 +τ dz ≤
n−2
j
Ωs ∩Bs |y−z|
j
(1+|z−X |)
1
Ωs ∩Bs |y−z|n−2
R
≤
1+|z−X s |
λ
τ −1
Cλ1−τ
n−2
(1+|y−X s |)min( 2 +2+1,n−2)
≤
C
1+|y−X s |
λ
where 0 < θ < min(2, n−2
− 1) := θ1 .
2
Similarly we also have
4
R
P
n−2
1
W
γ(z)
m
n−2
j
Ωs ∩Bs |y−z|
τ −1
1
(1+|z−X j |)
1
1
Ωs ∩Bs |y−z|n−2 (1+|z−X s |)2+ n+2
2 +τ
R
2
n+2
(1+|z−X s |)2+ 2 +τ
.
dz ≤
dz
C
(1+|y−X s |)
n−2 +τ
2
n−2 +τ +θ
2
dz ≤
C
(1+|y−X s |)
n−2 +τ +θ
2
,
MULTI-BUMP SOLUTIONS ON LATTICES
37
− τ )) := θ2 .
for 0 < θ < min(2, n−2
2
If y ∈ Ωi ∩ Bi for some i, from the above two inequalities, taking a
θ ∈ (0, min(θ1 , θ2 )), we have
4
1
Wmn−2 γ(z)
∪s Ωs ∩Bs |y−z|n−2
R
+C min
≤
1
s6=i
Cγ(y)
n−2 +τ +θ
2
(1+|y−X i |)
≤ Cγ(y)
P
(1+|y−X s |)
n−2 +τ
2
(1+|z−X j |)
n−2 +θ+1
2
,
dz ≤
Cγ(y)
(1+|y−X i |)
1
P
s6=i
(1+|y−X s |)
n−2 +τ +θ
2
by Lemma A.3
,
1
j
j
1
P
λτ −1
1
P
(1+|y−X j |)
n−2 +τ +θ
2
.
If y ∈ ∪i (Ωi ∩ Bic ), then γ(y) = 1 and it is easy to see that
4
1
Wmn−2 γ(z)
∪s (Ωs ∩Bs ) |y−z|n−2
R
≤
≤
C
4k
(λl) n−2
P
j
R
Rn
1
|y−z|n−2
C
(1+|y−X j |)
where 0 < θ < min(2 −
P
n−2 +τ +θ
2
4k n−2
, 2
n−2
j
P
j
1
(1+|z−X j |)
(1+|z−X j |)
n−2 +τ
2
dz
1
n−2 +τ +4− 4k
2
n−2
,
− τ ) := θ3 .
Thus we get
4
1
Wmn−2 γ(z)
∪i (Ωi ∩Bi ) |y−z|n−2
R
Cγ(y)
P
j
1
(1+|y−X j |)
for all 0 < θ < min(θ1 , θ2 , θ3 ).
n−2 +τ +θ
2
P
,
j
1
(1+|z−X j |)
n−2 +τ
2
dz ≤
n−2 +τ +θ
2
38
Y.Y. LI, J. WEI, AND H. XU
When z ∈ Ωi ∩ Bic , we estimate as follows: if y ∈ ∪i (Ωi ∩ Bic ), i.e.,
γ(y) = 1, we have
4
R
P
1
1
Wmn−2 γ(z) j
n−2 +τ dz
∪i (Ωi ∩B c ) |y−z|n−2
j
(1+|z−X |)
i
≤C
≤C
≤C
1
Rn |y−z|n−2
1
R
4 k
(λl) n−2
P
j
j
4
4+ n−2
2 +τ − n−2 k
(1+|z−X j |)
1
P

 j
(1+|y−X j |)
n−2 +τ
2
1
(1+|y−X j |)n−2
1
P
j
1
1
4k
4 k
(λl) n−2
when n ≥ 6
,
4k
(λl) n−2
when n = 5, k = 1,
,
(λl) n−2
n−2 +τ
2
(1+|y−X j |)
dz
1
4
min( n−2
2 +2+τ − n−2 k,n−2)
(1+|y−X j |)
 P

 j
≤ Cγ(y)
1
P
1
2
1
.
4k
(λl) n−2
4
+ 2 + τ − n−2
k < n − 2 which
Here in the case n ≥ 6, we need n−2
2
4
n−2
n−2
.
gives τ < 2 − 2 + n−2 k = τ0 . Notice that when k < 2 , k < τ0 < n−2
2
Therefore the set for τ is not empty when n ≥ 6. When n = 5, k = 1
4
since k < n−2
. In this case n − 2 < n−2
+ 2 + τ − n−2
k and we can just
2
2
n−2
choose k ≤ τ < τ0 = 2 .
When y ∈ Ωi ∩ Bi for some i,
4
R
P
n−2
1
W
γ(z)
m
c
n−2
s
∪j (Ωj ∩B ) |y−z|
j
≤C
1
≤ C λτ1−1
≤
≤
1
1
Rn |y−z|n−2 λτ −1
R
4 k
(λl) n−2
C
λτ −1
P
j
P
j
1
1
(1+|z−X s |)
1
4
4+ n−2
2 +1− n−2 k
4
min( n−2
2 +2+1− n−2 k,n−2)
(1+|y−X j |)
1
C
4k
(λl) n−2
≤ Cγ(y)



P
j
γ(y)
(1+|y−X i |)
n−2 +τ
2
,
γ(y)
(1+|y−X i |)n−2+τ −1
1
(1+|y−X j |)
n−2 +τ
2
1
1
1
4
(λl) n−2
4 k
(λl) n−2
if n − 2 >
≤
dz
(1+|z−X s |)
4
min( n−2
2 +2+1− n−2 k,n−2)
(1+|y−X i |)




n−2 +τ
2
+3−
γ(y)
(1+|y−X i |)
4
(λl) n−2
k
.
k
due to y ∈ Ωi ∩ Bi
,
n−2
2
dz
n−2 +τ
2
4
n−2
if n − 2 <
n−2
2
+3−
4
n−2
MULTI-BUMP SOLUTIONS ON LATTICES
39
4
+ 2 + 1 − n−2
k = n − 2, the log |y| term from applying Lemma
When n−2
2
A.2 will be harmless as long as we choose τ < τ0 ≤ n−2
. The fact that
2
4k
2 > n−2 is also used in the above. Combining the above together, for
0 < θ < min(θ1 , θ2 , θ3 ), we conclude that
4
R
P
n−2
1
1
γ(z) j
W
m
n−2 +τ dz ≤
Rn |y−z|n−2
j
(1+|z−X |)
P
Cγ(y)
j
(1+|y−X j |)
n−2 +θ+τ
2
+
Cγ(y)
4 k
(λl) n−2
2
1
P
j
(1+|y−X j |)
n−2 +τ
2
.
C
. If kφk∗ ≤ n+2
, then
Lemma A.5. Assume n ≥ 4 and 0 < τ < n+2
2
λ 2 −τ
for any c > 0, there exists a constant λ0 = λ0 (n, k, τ, C, c) > 0, such that
for any λ > λ0 , φ(y) ≤ cWm (y) in ∪i (Bi ∩ Ωi ).
Proof. We prove by contradiction. Without loss of generality, we may
assume that φ(y) ≥ cWm (y) for some y ∈ B1 ∩ Ω1 . Note that γ(y) ≤ 1.
By Lemma A.3, we have
P
1
1
φ(y) ≥ cC j (1+|y−X
j |)n−2 ≥ C (1+|y−X 1 |)n−2
≥C
≥ Cγ(y)
P
j
When n ≥ 4 and 0 < τ <
n−2 +τ
2
1
(1+|y−X j |)
n−2
,
2
C
λ
1
1
(1+|y−X 1 |)
n+2
−τ
2
(1+|y−X 1 |)
n−2 +τ
2
n−2 −τ
2
1
(1+|y−X 1 |)
n−2 −τ
2
.
we have
≥ kφk∗ ≥
1
(λl)
n−2
−τ
2
.
This gives a contradiction when λ is large. In the case n ≥ 4 and
> τ ≥ n−2
, noting the fact that
2
n+2
2
1
we get
C
λ
n+2 −τ
2
(1 + |y − X 1 |)
n−2
−τ
2
≥ 1,
∀y ∈ B1 ,
≥ kφk∗ ≥ C, which is impossible when λ large.
Lemma A.6. For any φ ∈ M̃, we have, for some C > 0, independent of
m and l,
Z
4
Ckφk∗
|
Kλ (z)Wmn−2 φZs,tdz| ≤ n−2 ,
λ 2 +τ
Rn
40
Y.Y. LI, J. WEI, AND H. XU
and
|
Z
4
Rn
Kλ (z)Wmn−2 φσi dz| ≤
Ckφk∗
λ
n−2
+τ
2
.
Proof. By the orthogonality condition
4
4
R
R
n−2
n−2
K
(z)W
φZ
dz
=
(K
(z)
−
1)σ
Zs,t φdz
m
s
λ
s,t
λ
Rn
Rn
+O(1)|
4
R
Rn
n−2
Zs,tφdz| + O(1)|
Ŵm,s
4
R
Rn
Ŵm σsn−2
−1
Zs,t φdz|.
Using Lemma A.3 and the proof of Lemma A.4 in Ωi with i 6= s, we get
4
R
R
P
γ(z)
n−2
1
| Ωi ∩Bi Ŵm,s
Zs,t φdz| ≤ Ckφk∗ Ωi ∩Bi j
n−2 +τ (1+|z−X s |)n−2 ×
j
(1+|z−X |)
P
1
k6=s (1+|z−X k |)n−2
≤
Ckφk∗
λτ −1
≤
λτ −1 |X i −X s | 2
4
n−2
2
dz
1
1
dz
(1+|z−X s |)n−2
Ωi ∩Bi (1+|z−X i |) n−2
2 +5
R
Ckφk∗
n
.
When z ∈ ∪i (Ωi ∩ Bic ), we use the same idea as in the proof for Lemma
A.4 to get
4
R
n−2
Zs,t φdz|
| ∪i (Ωi ∩Bc ) Ŵm,s
i
≤
Ckφk∗
4 k
(λl) n−2
Ckφk∗
R
∪i (Ωi ∩Bic )
P
≤
(λl) n−2
≤
λτ −1 (λl) 2
4
k
Ckφk∗
n
P
j
1
(1+|z−X j |)
n−2 +τ +4− 4 k
2
n−2
1
j6=s
1
dz
(1+|z−X s |)n−2
4
min(n−2, n−2
2 +τ +2− n−2 k)
(1+|X j −X s |)
+
1
4
2+ n−2
2 +τ − n−2 k
(λl)
.
For i = s, note that for any z ∈ Ωs , taking X j be the closest point in
Xl,m to X s (there are at most 2k such kind of points in Xl,m ), we have
4
n−2
Ŵm,s
≤
C
By Lemma A.1 we also get
Z
4
n−2
Zs,tφdz| ≤
Ŵm,s
|
Ωs ∩Bs
4
4
(1 + |z − X j |)4− n−2 k (λl) n−2 k
Ckφk∗
.
n
(λl) 2 λτ −1
.
MULTI-BUMP SOLUTIONS ON LATTICES
Hence we deduce that
Z
4
n−2
|
Zs,tφdz| ≤
Ŵm,s
Rn
41
Ckφk∗
n
(λl) 2 λτ −1
for all n ≥ 5 and k ≤ τ < τ0 .
Similarly we have
4
4
R
R
−1
| Rn Ŵm,s σsn−2 Zs,t φdz| ≤ C Rn Ŵm,s σsn−2 |φ|dz
≤
|
R
Rn
Ckφk∗
.
n
(λl) 2 λτ −1
4
(Kλ (z) − 1)σsn−2 Zs,t φdz|
≤ Ckφk∗
γ(z)
|Kλ (z) − 1| (1+|z−X
s |)n+2
R
Rn
P
j
1
(1+|z−X j |)
n−2 +τ
2
dz.
It is easy to see by Lemma A.1 and the relations between X i that
Z
1
|Kλ (z) − 1|γ(z) X
C
|
dz|
≤
.
n−2
n−2
s
n+2
+τ
j |) 2 +τ
2
(1
+
|z
−
X
(λl)
Rn (1 + |z − X |)
j6=s
So we only need to estimate the integral when j = s. Let Ω = {z ∈
R ||z − X s | ≤ λ. By Lemma A.3 and integrating, it is not hard to get
R
γ(z)
1
|Kλ (z) − 1| (1+|z−X
n−2 +τ dz
s |)n+2
Ω
s
n
(1+|z−X |)
C
λβ+τ −1
≤
≤C
R
Ωc


λ
|z−X s |β+τ −1
Ω (1+|z−X s |)n+2+ n−2
2 +τ
1
if β >
n+2 +τ
2
1
λβ+τ −1
≤
1
λ
n−2 +τ
2
n
2
dz
+ 2,
if β ≤
γ(z)
|Kλ (z) − 1| (1+|z−X
s |)n+2
≤C
≤



R
2
n
2
+ 2,
1
(1+|z−X s |)
n−2 +τ
2
dz
1
dz
Ωc (1+|z−X s |)n+2+ n−2
2 +τ
R
C
n+2
λ 2 +τ
.
since β > n − 2 > n2 when n ≥ 5.
Therefore the first inequality can be derived easily and the second one
can be proved similarly.
42
Y.Y. LI, J. WEI, AND H. XU
Lemma A.7. Under the assumption of Lemma A.4, for any h ∈ D̃ and
4
φ ∈ M̃, let φ̃ = PE (−∆)−1 (h + Kλ Wmn−2 φ), then there exist an integer l0
and a constant C > 0, depending only on K, n, β, τ , C1 and C2 , such
that for any l ≥ l0 , we have
kφ̃k∗ ≤ C(khk∗∗ + kφk∗).
Proof. By assumption on φ, φ satisfies the equation
4
1
φ̃(y) =
n(n − 2)ωn
Z
Rn
X
X
h + Kλ Wmn−2 φ
dz
+
c
Z
+
bi σi ,
i,j
i,j
|y − z|n−2
i,j
i
for some constants ci,j , bi .
We first claim that, for some constant C, independent of m and l,
1
C
.
(A6)
|ci,j |, |bi | ≤ Ckhk∗∗ + n kφk∗
τ
2
λ −1
λ
4
In fact, multiplying σsn−2 Zs,t on both side of the equation and integrating, we get
!
Z
n+2
X
X
4
bi σin−2 Zs,t dz,
(A7)
ci,j hZi,j , Zs,t i =
−h − Kλ Wmn−2 φ −
Rn
i,j
|
R
Rn
h(z)Zs,t dz| ≤ Ckhk∗∗
≤ Ckhk∗∗
+
where
P
j6=s
R
Rn
j
1
γ(z)
(1+|z−X s |)n−2
R
Rn
R
γ(z)
Rn (1+|z−X s |)n−2+ n+2
2 +τ
+
≤
1
Bsc (1+|z−X s |)n−2+ n+2
2 +τ
R
C
λτ −1
+
C
(λl)
n−2 +τ
2
j
1
(1+|z−X j |)
≤
≤
dz
C
,
λτ −1
n+2 +τ
2
dz
γ(z)
1
dz
(1+|z−X s |)n−2 (1+|z−X j |) n+2
2 +τ
γ(z)
dz
Rn (1+|z−X s |)n−2+ n+2
2 +τ
R
P
,
1
1
dz
Bs λτ −1 (1+|z−X s |)n−2+ n+2
2 +1
R
dz
MULTI-BUMP SOLUTIONS ON LATTICES
43
and
P
j6=s
γ(z)
1
dz
Rn (1+|z−X s |)n−2 (1+|z−X j |) n+2
2 +τ
R
≤
1
λτ −1
≤
λτ −1
≤
C
.
λτ −1
1
P
j6=s
P
j6=s
1
1
dz
Rn (1+|z−X s |)n−2−τ +1 (1+|z−X j |) n+2
2 +τ
R
1
n−2
|X s −X j | 2
(
1
Rn (1+|z−X j |)n+1
R
Here we have used the fact that
n−2
.
2
P
j6=s
+
1
|X j −X s |
1
dz)
(1+|z−X s |)n+1
n−2
2
converges when 1 ≤
Thus we have derived
Z
C
|
h(z)Zs,t dz| ≤ τ −1 khk∗∗ .
λ
Rn
By Lemma A.6,
Z
4
Ckφk∗
|
Kλ (z)Wmn−2 φZs,tdz| ≤ n−2 +τ ,
λ 2
Rn
By Lemma A.1 and symmetry of σi , it is easy to check that,
k<
hZi,j , Zs,ti = 0 if i = s and j 6= t;
hZi,j , Zi,j i = C;
and
C
|hZi,j , Zs,ti| ≤
, if i 6= s,
|X i − X s |n−2
Z
n+2
σin−2 Zs,t = 0, if i = s;
Rn
and
Z
C
, if i 6= s.
− X s |n−2
Rn
Notice that the left hand of equation (A7) can be viewed as a linear
system with variables ci,j of (m + 1)k (n + 1) dimension and coefficient
matrix of (m + 1)k (n + 1) × (m+ 1)k (n + 1) with entry hZi,j , Zs,t i. If we
denote G = (hZi,j , Zs,t i) = as,t
i,j be this matrix and let X = (xs,t ) be in
(m+1)k (n+1)
R
with maximum norm denotes as |X| = maxs,t |xs,t |, then
P s,t
c(n+1)
C|xi,j | + (λl)
n−2 |X| ≥ |
s,t ai,j xs,t |
|
n+2
σin−2 Zs,t| ≤
= |Cxi,j +
P
(s,t)6=(i,j)
|X i
as,t
i,j xs,t | ≥ C|xi,j | −
c(n+1)
|X|,
(λl)n−2
44
Y.Y. LI, J. WEI, AND H. XU
1
dz
Rk 1+|z|n−2
R
where c is controlled by
implies that
and doesn’t depend on m. This
C
|X|
2
with C independent of m when λ is large enough. Therefore, we obtain
that
C
1
1
|ci,j | ≤ Ckhk∗∗ + n kφk∗
+ C max |bi |
.
τ
−1
λ
(λl)n−2
λ2
where C is a constant that doesn’t depend on m and l.
Similar estimates for bi can be obtained in the same way as ci,j , we skip
the detail. Thus we prove (A6).
2C|X| ≥ |GX| ≥
It follows from Lemma A.2,
R
R
| Rn |y−z|1 n−2 h(z)dz| ≤ Ckhk∗∗ Rn
P
≤ Ckhk∗∗ γ(y)
j
and
|
≤C
R
Rn
1
P
j
(1+|y−X j |)
1
(1+|z−X j |)
n−2 +τ
2
n+2 +τ
2
dz
,
4
1
σ n−2 Zi,j dz|
|y−z|n−2 i
1
1
dz
Rn |y−z|n−2 (1+|z−X i |)n+2
R
γ(z)
|y−z|n−2
≤
C
(1+|y−X i |)
n−2 +τ
2
∀k ≤ τ < τ0 .
,
Similarly,
|
Z
Rn
n+2
1
C
n−2
σ
dz| ≤
n−2
i
n−2
|y − z|
(1 + |y − X i |) 2 +τ
when k ≤ τ < τ0 .
This, combined with Lemma A.4 and (A6), gives the conclusion. The
proof of Lemma is thus completed.
Lemma A.8. Under the same assumption of Lemma A.4, there exist an
integer l0 and a constant C ≥ 1, depending only on K, n, β, τ , C1 and
C2 , such that for any φ ∈ Ẽ, we have
4
n+2
kφk∗
−1
n−2
(A8)
kφ −
Kλ Wm φ k∗ ≥
PE (−∆)
.
n−2
C
Proof. (A8) is equivalent to
φ(y) = h +
(A9)
+
P
i bi σi
+
n+2
n(n−2)2 ωn
P
R
Rn
i,j ci,j Zi,j ,
4
1
K (z)Wmn−2 (z)φ(z)dz
|z−y|n−2 λ
MULTI-BUMP SOLUTIONS ON LATTICES
45
for some h ∈ Ẽ and constants bi , ci,j . Then by Lemma A.4 and the proof
of Lemma A.7, we get
−1
P
1
∗
γ(y) X i ∈Xl,m
|φ(y) ≤ khk∗ + Ckφk
n−2 +τ
n−2 +τ
i
(1+|y−X |)
(A10)
+Ckφk∗
1
4k
(λl) n−2
+
P
λ
2
1
X j ∈Xl,m
n−2 +τ +θ
(1+|y−X j |) 2
P
1
n−2 +τ
X j ∈Xl,m
(1+|y−X j |) 2
!
2
.
We show that kφk∗ ≤ Ckhk∗ for l large enough. If not, we can find
sequences l → ∞, ml ≥ 1, Λi,l ∈ [C1 , C2], P i,l ∈ B 1 (X i ), bi,l , ci,j,l and φl
2
with kφl k∗ = 1, such that (A9) is satisfied for khl k∗ → 0. We may assume
that kφl k∗ = 1. Therefore for some y l ∈ Rn , we obtain from (A10) that


P
1
(A11)
X j ∈Xl,m
1 = kφl k∗ ≤ C khl k∗ + P
(1+|y l −X j |)
1
X j ∈Xl,m
n−2 +τ +θ
2
n−2
(1+|y l −X j |) 2 +τ
.
Then there is a R > 0 independent of m, such that for some i(l),
y l ∈ BR (X i(l) ) for all l large. (If y l is far away from all X i , the right side
of (A11) is approaching zero as l → ∞.) Hence we get that
max |λτ −1 φl (y)| ≥ a > 0.
BR (xi(l) )
From the proof of Lemma A.3, for any fixed R > 0, it is easy to see that
Wm (x − P i(l) ) → σ0,Λ for some Λ ∈ [C1 , C2] in BR as l → ∞ independent
of m. Multiplying λτ −1 on both side of the equation (A9) and using the
estimates (A6) for bi,l and ci,j,l and the fact that khl k∗ → 0, we can see
that φ̃l (y) := λτ −1 φ(y − P i(l) ) converges uniformly in any compact set to
a non-zero solution φ̃ of
(A12)
−∆φ̃ −
4
n + 2 n−2
σ0,Λ φ̃ = 0
n−2
for some Λ ∈ [C1 , C2 ]. We will show that φ̃ is perpendicular to the kernel
of (A12) and therefore φ̃ = 0, which is a contradiction.
For this purpose, by Lemma A.3, we get

1
y ∈ Bj ∩ Ωj

 (1+|y−X j |) n−2
2 +1
λτ −1 φ(y) ≤ Φ(y) = C

1
 λτ −1k
y ∈ B c ∩ Ωj .
n−2
(λl) (1+|y−X j |)
2
+τ −k
j
46
Y.Y. LI, J. WEI, AND H. XU
Since
|
σi (y),
∂σi
(y)|,
∂P i j
|
∂σi
C
(y)| ≤
,
∂Λi
(1 + |y − X i |)n−2
from dominated convergence theorem and the fact that hΦ, σi i = 0,
∂σi
i = 0, we obtain
0 and hΦ, ∂Λ
i
hφ̃, σ0,Λ i,
hφ̃,
∂σ0,Λ
i,
∂xj
hφ̃,
∂σi
hΦ, ∂P
i i =
j
∂σ0,Λ
i = 0.
∂Λ
Therefore we have proved the conclusion.
Lemma A.9. For j = 1, ..., n, i = 1, ..., (m + 1)k , 0 < C1 ≤ Λi ≤ C2 < ∞
and P i ∈ B 1 (X i ), we have
2
R
Rn
n+2
∂σi
Kλ (y)σin−2 ∂P
i =
j
+O( |P
i −X i |2
λβ
Dn,β aj
Λβ−2
λβ
i
(P i j − X i j )
) + o( λ1β ).
R
n
|x |β
where Dn,β = (n(n − 2)) 2 (n − 2)β Rn (1+|x|j2 )n+1 dx. oλ (1) only depends on
the condition of function R( λy ) near X i and o(1) → 0 as l → ∞ (or same
as λ → ∞) (see the remark 5.1 below).
MULTI-BUMP SOLUTIONS ON LATTICES
47
β−n
Proof. Let δ = λ 2n . We have
n+2
2n
R
R
2
i
n−2 ∂σi
n−2 Λi (yj −P j )
K
(y)σ
=
(n
−
2)
K
(y)σ
dy
2
λ
λ
n
i
n
i
i
∂P j
R
R
1+Λ |y−P i |2
i
= (n − 2)
=
n−2
λβ
2n
R
2n
R
i
P
|y−X i |≤δλ
h
Λ2 (y −P i )
Λ2 (y −P i )
j
1
i j
Kλ (y)σin−2 1+Λ
2 |y−P i |2 dy + O( (δλ)n+1 )
|y−X i |≤δλ
ah |yh − X i h |β + o(1)|y − X i |β
j
1
i j
×σin−2 1+Λ
2 |y−P i |2 dy + O( (δλ)n+1 )
i
=
n−2
λβ
2n
R
|y−X i |≤δλ
+o( λ1β ) + o( |P
i
i −X i |β
)
λβ
n
= (n(n − 2)) 2 n−2
λβ
R
Rn
+O(|P i − X i |2 )) ×
(n−2)βaj
Λβ−2
λβ
i
+o( λ1β ) + O( |P
i −X i |2
λβ
P
h
ah (|xh |β + β|xh |β−2 xh (P i − X i )h
Λ2i xj
Λn
i
dx
2
2
n
(1+Λi |x| ) (1+Λ2i |x|2 )
= (n(n − 2)) 2
n
Λ2 (y −P i )
j
i β n−2 i j
dy
h ah |yh − X h | σi
1+Λ2 |y−P i |2
P
|xj |β
(P i j
Rn (1+|x|2 )n+1
R
) + o( |P
i −X i |β
λβ
n
If we let Dn,β = (n(n − 2)) 2 (n − 2)
proof.
).
R
Rn
+ o( λ1β ) + o( |P
i −X i |β
λβ
)
− X ij )
|xj |β
dx,
(1+|x|2 )n+1
we complete the
Remark 5.1. The oλ (1) only depends on the condition of R( λy ) near X i ,
the estimates doesn’t depend on Pi as long as |Pi − X i | ≤ 21 . If we know
more, say |∇R(x) ≤ C|x|β−1+s near 0 for some small s > 0, then oλ (1) =
C
.
λs
Lemma A.10.
R
n+2
∂σi
Kλ (y)σin−2 ∂Λ
=
Rn
i
i
i β−1
C3
Λβ+1
λβ
i
|
) + o( λ1β ).
+O( |P −X
λβ
where o(1) is the same as in Lemma A.9 and
Z
n
β[n(n − 2)] 2 (n − 2) X
|y1 |β
C3 = −
(
ai )
dy > 0.
2
2n
Rn (1 + |y| )
i
48
Y.Y. LI, J. WEI, AND H. XU
Proof. Observe that
n+2
R
∂σi
=
Kλ (y)σin−2 ∂Λ
i
n
2 (n−2)
= − Λ1i [n(n−2)]2n
n
2 (n−2)
=
C3
Λβ+1
λβ
i
R
|y|≤δλ
= − Λβi [n(n−2)]2n
+o( λ1β ) + O( |P
n−2 ∂
2n ∂Λi
R
Rn
i −X i |β−1
λβ
)
Kλ (y)σin−2
y
(∇K( λΛ
+
i
P
+ o( λ1β ) + O( |P
2n
R
Pi
)
λ
·
y
) 1 dy
λΛi (1+|y|2 )n
1
+ O( (δλ)
n)
β
1
ah (Λ|yihλ)| β (1+|y|
2 )n dy+
i −X i |β−1
λβ
).
Lemma A.11. For j = 1, ..., n, we have
n+2
R
R
n+2
Kλ (y)(W̄m + φ) n−2 Zi,j dy = Rn Kλ (y)σin−2 Zi,j dy
Rn
n+2
+C |ǫ|2 + kφk∗n−2
1
n +τ n+2
n−2
(λl) 2
+
kφk2∗ λ2τ1−2
+
1
λβ+1
+
o( λ1n )
.
Proof. We begin with
n+2
4
n+2
|(W̄m + φ) n−2 − Wmn−2 −
≤C

n+2

 |φ + ǫWm | n−2 ,


n+2
Wmn−2 (ǫWm
n−2
+ φ)|
if |φ| ≥ Wm ;
6−n
Wmn−2 |ǫWm + φ|2 ,
if |φ| ≤ Wm .
By Lemma A.5, when λ is large, if |φ| ≥ Wm , then y ∈ (∪j (Bj ∩ Ωj ))c :=
Ω. So
R
n+2
Kλ (y)(W̄m + φ) n−2 Zi,j dy
Rn
n+2
= Rn Kλ (y) Wmn−2 +
R
+O(1)
R
n+2
4
n+2
Wmn−2 (ǫWm
n−2
|φ + ǫWm | n−2 |Zi,j | + O(1)
Ω
R
+ φ) Zi,j dy
Rn
6−n
Wmn−2 |ǫWm + φ|2 |Zi,j |.
By Lemma A.1, Lemma A.2 and similar argument as in the proof of
Lemma A.4, we get easily that
MULTI-BUMP SOLUTIONS ON LATTICES
n+2
R
≤
4 k
(λl) n−2
1
R P
j
Ω
|φ| n−2 |Zi,j |dy
Ω
n+2
Ckφk∗n−2
n+2
R
|φ + ǫWm | n−2 |Zi,j | ≤ C
Ω
(1+|z−X j |)
49
n+2 + n+2 τ − 4 k
2
n−2
n−2
1
dz
(1+|z−X i |)n−1
n+2
Ckφk∗n−2
≤
n +τ n+2
n−2
by Lemma A.1.
,
(λl) 2
Using Lemma A.1, Lemma A.2 and Lemma A.3, we can infer that
R
6−n
Wmn−2 |ǫWm + φ|2 |Zi,j |dy ≤
Rn
C
R
Rn
n+2
n−2
|ǫ|2 Wm
2
≤ C |ǫ| +
4
n−2
|Wm
4
n−2
− σi
|≤C
+ Wm |φ|2 |Zi,j |dy
kφk∗
λτ −1





6−n
n−2
2 .
4
n−2
Ŵm,i
,
4
σin−2
−1
if Ŵm,i > σi ;
if Ŵm,i ≤ σi .
Ŵm,i ,
By Lemma A.3, we know that Ŵm,i ≤ σi in Bi (we may need to shrink
the ball a little bit) and Ŵm,i ≥ σi in each Bj with j 6= i. Since hφ, Zi,j i =
0, we can get,
R
4
Kλ (y)Wmn−2 φZi,j dy =
Rn
=
≤
4
R
Rn
4
Kλ (y)σin−2 φZi,j dy + o(1) λ1n
(Kλ (y) − 1)σin−2 φZi,j dy + o( λ1n )
Rn
R
C
λβ+1
+ o( λ1n ).
Similarly
R
Rn
4
Kλ (y)Wmn−2 ǫWm Zi,j dy =
C|ǫ|
1
+ (λl)
n−2 ≤ o( λn ).
R
Rn
n+2
(Kλ (y) − 1)ǫi σin−2 Zi,j dy
50
Y.Y. LI, J. WEI, AND H. XU
For n ≥ 5 it holds that
n+2
n+2
≤C

n+2
n−2

,
 Ŵm,i


4
n+2 n−2
σ Ŵm,i |
n−2 i
|Wmn−2 − σin−2 −
if Ŵm,i ≥ σi ;
4
2
Ŵm,i
σin−2
−1
n
n
n−2
σin−2
≤ Ŵm,i
−1
,
Using Lemma A.1, we deduce that
Z
n+2
n−2
|Kλ (y)Ŵm,i
Zi,j |dy ≤
Rn
and
Z
n
Rn
n
n−2
|Kλ (y)Ŵm,i
σin−1
−1
when Ŵm,i ≤ σi .
C
,
(λl)n−1
Zi,j |dy ≤
C
.
(λl)n−1
For s 6= i, by change of variable for s 6= i,
4
n+2
R
R
n−2
n−2
n+2
∂
K
(y)σ
K
(y)σ
σ
Z
dy
=
σs dy
λ
λ
s
i,j
i
n
n
i
i
n−2 R
∂P j R
=
∂
∂P i j
=
1
λ
R
R
Rn
n+2
i
Rn
n−2
σ s i dt
K( t+P
)σ0,Λ
λ
i P −P ,Λs
i
n+2
∂K( t+P
) n−2
λ
σ0,Λi σP s −P i ,Λs dt
∂tj
−
i
n+2
n−2
)σ0,Λ
K( t+P
λ
Rn
i
R
∂σP s −P i ,Λs
dt.
∂P s j
From the above, using Lemma A.1, it is easy to see that
Z
4
C
C
|
Kλ (y)σin−2 Ŵm,i Zi,j dy| ≤
+
.
λ(λl)n−2 (λl)n−1
Rn
Similarly
X
∂σi
C(1 + |ǫ|)
(1 + ǫs )h i , σs i| ≤
|
.
∂P j
(λl)n−1
s6=i
The above estimates give the conclusion.
Lemma A.12. For some constant C > 0 independent of i, j and m,
Z
Z
n+2
n+2
∂σj
∂σj
C
n−2
dx −
σin−2
|≤ i
.
|
Kλ (x)σi
∂Λj
∂Λj
|P − P j |n−2 λ2
Rn
Rn
Proof. Take a δ > 0 small, such that |K(x) − 1| ≤ c|x|β for some c > 0
and x ∈ Bδ (0). Since K(X i ) = 1, we get
n+2
n+2
R
R
n−2 ∂σj
n−2 ∂σj
dx
=
σ
+
K
(x)σ
λ
n
n
i
i
∂Λj
R
∂Λj
R
R
c (X i )∩B c (X j ))
Bδλ (X i )∪Bδλ (X j )∪(Bδλ
δλ
n+2
∂σ
(Kλ (x) − 1)σin−2 ∂Λjj dx,
MULTI-BUMP SOLUTIONS ON LATTICES
and also
n+2
R
R
∂σ
| Bδλ (X i ) (Kλ (x) − 1)σin−2 ∂Λjj dx| ≤ C Bδλ (X i )
≤C
β
R
Bδλ (0)
+
( |x|
λβ
C
|P i −P j |n−2
≤
C
(1
|P i −P j |n−2 λ2
R
Bδλ (0)
+
n+2
|x−X i |β n−2
σi σj dx
λβ
|P i −X i |β
1
1
)
n+2
n−2
λβ
(1+|x|2 ) 2 (1+|x−P j +P i |2 ) 2
β
≤
51
+
( |x|
λβ
1
)
λβ
|P i −X i |β
1
)
n+2
λβ
(1+|x|2 ) 2
dx
dx
C
.
|P i −P j |n−2 λ2
≤
Similarly,
|
n+2
R
Bδλ
|
(A13)
∂σ
(Kλ (x) − 1)σin−2 ∂Λjj dx|
C
(λ2
|P i −P j |n+2
≤
Lastly,
(X j )
+ λ3−β ) ≤
C
.
|P i −P j |n−2 λ2
n+2
R
c (X i )∩B c (X j )
Bδλ
δλ
≤C
≤C
∂σ
(Kλ (x) − 1)σin−2 ∂Λjj dx|
n+2
R
c (X i )∩B c (X j )
Bδλ
δλ
σin−2 σj dx
1
c (0)∩B c (X j −X i )
n+2
n−2
Bδλ
δλ
(1+|x|2 ) 2 (1+|x−P j +P i |2 ) 2
R
dx
Let z = P j − P i and 2d = |z|. To estimate (A13), we will use the
method used by Wei-Yan in [49]. Since d > δλ when l is large, we can
split
c
c
Bδλ
(0) ∩ Bδλ
(X j − X i ) = A1 ∪ A2 ∪ A3 ,
where A1 = Bd (0) \ Bδλ (0), A2 = Bd (X j − X i ) \ Bδλ (X j − X i ) and
A3 = Bdc (0) ∩ Bdc (X j − X i ).
R
R
1
C
1
n−2 dx ≤ dn−2
n+2
n+2 dx
A1
A1
2
2
2
(1+|x| )
≤
2
(1+|x−z| )
(1+|x| )
2
2
C
.
|P i −P j |n−2 λ2
Similarly it holds that
Z
A2
(1 +
On A3 , from [49],
|x|2 )
1
n+2
2
(1 + |x −
1
(1 + |x|2 )
n+2
2
(1 + |x − z|2 )
n−2
2
z|2 )
≤
n−2
2
dx ≤
|P i
C
.
− P j |n
C
C
≤ 2n ,
n+2
+ |x|)
|x|
|x|n−2 (1
52
Y.Y. LI, J. WEI, AND H. XU
therefore, we infer that
Z
A3
(1 +
which gives
Z
|
1
|x|2 )
n+2
2
z|2 )
(1 + |x −
n+2
c (X i )∩B c (X j )
Bδλ
δλ
(Kλ (x) − 1)σin−2
n−2
2
dx ≤
|P i
C
,
− P j |n
C
∂σj
dx| ≤ i
.
∂Λj
|P − P j |n−2 λ2
The conclusion of Lemma follows easily.
Lemma A.13. For some constant C > 0 independent of i, j, m,
Z
Z
4
n+2
n−2
C
∂σj
∂σj
n−2
|
Kλ (x)σj σi
σin−2
dx −
dx| ≤ i
.
∂Λj
n + 2 Rn
∂Λj
|P − P j |n−2 λ2
Rn
Proof.
R
4
n−2
Rn
Kλ (x)σj
n+2
∂σ
σi ∂Λjj dx
=
n−2
n+2
n+2
=
n−2
n+2
R
=
n−2
n+2
R
Rn
Rn
∂σjn−2
∂Λj
n+2
n−2
σi
σi dx +
∂σj
dx
∂Λj
n−2
n+2
R
n−2
n+2
R
+
∂σjn−2
K
(x)
λ
n
R
∂Λj
R
Rn
σi dx
n+2
(Kλ (x) −
∂σjn−2
1) ∂Λ
j
σi dx
n+2
Rn
(Kλ (x) −
∂σjn−2
1) ∂Λ
j
σi dx.
The second error term in the above can be similarly estimated as in
Lemma A.12 and the proof is thus completed.
6. Appendix B
We recall some results proved in [34], in a form convenient for our
application.
Let {Ki } be a sequence of functions satisfying, for some constant A ≥ 1,
1
(B1)
≤ Ki ≤ A, in B2 ∀i.
A
where B2 is the ball in Rn of radius 2 centered at the origin.
For β > n − 2, recall that {Ki } satisfies (∗)β for some positive constants
L1 and L2 (independent of i) in B2 if {Ki } ⊂ C [β]−1,1(B2 ) satisfies
and, if β ≥ 2, that
|∇Ki | ≤ L1 ,
β−s
|∇s Ki (y)| ≤ L2 |∇Ki (y)| β−1 ,
in B2 ,
for all 2 ≤ s ≤ [β],
y ∈ B2 .
MULTI-BUMP SOLUTIONS ON LATTICES
53
Remark 6.1. Conditions (H1), (H2) and (H3) guarantee that K satisfies (∗)β in a neighborhood of 0.
Proposition B.1. For n ≥ 3, A ≥ 1, L1 , L2 > 0 and β > n − 2, let
{Ki } be a sequence of functions satisfying (B1) and (∗)β for L1 and L2 .
Let {ui } be a sequence of C 2 solutions of
n+2
−∆ui = Ki uin−2 , ui > 0, B2 ,
satisfying, for some constant a independent of i,
kui k
2n
L n−2 (B2 )
≤ a < ∞,
∀i.
Then, after passing to a subsequence, {ui } either stays bounded in B1 , or
has only isolated simple blow up points in B1 .
See Definition 0.3 in [34] for the definition of isolated simple blow up
points.
Proof. It is easy to see, using the equation of ui , that
k∇ui kL2 (B 3 ) ≤ C(n, A, a) := Ca .
2
Let δ0 > 0 be the small constant given in Proposition 2.1 in [32], and fix
a positive integer k such that
2n
For rl = 1 +
Since
Pk R
l=1
l−1
,
2k
Ca2 + a n−2 ≤ δ0 k,
1 ≤ l ≤ k + 1, let
Al = {x|rl ≤ |x| ≤ rl+1 },
n+2
(|∇ui |2 + uin−2 ) ≤
Al
1 ≤ l ≤ k.
n+2
2n
(|∇ui |2 + uin−2 ) ≤ Ca2 + a n−2 ≤ δ0 k,
B3
R
2
there exist some 1 ≤ l ≤ k and a subsequence of {ui } (still denoted as
{ui }) such that
Z
n+2
(|∇ui |2 + uin−2 ) ≤ δ0 , ∀i.
Al
It follows from Proposition 2.1 in [32] that
kui kL∞ (Âl ) ≤ C(δ0 , n, A, a),
1
1
where Âl = {x|rl + 8k
< |x| < rl+1 − 8k
}. Using this estimate, we work on
the ball Brl+1 − 1 . Then the proofs of Proposition 4.1, Proposition 4.2 and
8k
Theorem 4.2 in [34] apply, in view of the fact that {ui } stays bounded in
the shell Âl . We obtain the conclusion.
54
Y.Y. LI, J. WEI, AND H. XU
References
[1] A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic
problems on Rn . Progress in Mathematics Series, vol. 240 (Birkhäuser, 2006).
[2] W.Ao, J. Wei and J. Zeng, An optimal bound on the number of interior spike
solutions for the Lin-Ni-Takagi problem . J. Funct. Anal. 26 (2013), no. 7, 13241356.
[3] G. Arioli, Andrzej Szulkin and Wenming Zou, Multibump solutions and critical
groups. Trans. Amer. Math. Soc. 361 (2009), 3159 - 3159,
[4] T. Aubin, Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
[5] A. Bahri, Critical points at infinity in some variational problems. Pitman Research notes in Mathematics 182, 1982.
[6] A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical
Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl.
Math. 41 (1988), 253 - 294
[7] A. Bahri and J.M. Coron, The scalar curvature problem on stardard three dimensional sphere. J. Funct. Anal. 95 (1991), 106 - 172.
[8] A. Bahri, Y.Y. Li and O. Rey, On a variational problem with lack of compactness:
the topological effect of the critical points at infity. Calc. Var. Partial Differential
Equations 3 (1995), 67 - 93.
[9] H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent–
survey and perspectives. Directions in partial differential equations (Madison,
WI, 1985), 17-36, Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press,
Boston, MA, 1987.
[10] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations
involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983) 437 477.
[11] H. Brezis and L. Nirenberg, Nonlinear Functional Analysis. Book in preparation.
[12] Jaeyoung Byeon and Kazunaga Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schroedinger equations. Memoirs of the American
Mathematical Society 229 (2014), No. 1076.
[13] Jaeyoung Byeon and Kazunaga Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential,
Journal of the European Mathematical Society 15 (2013), no. 5, 1859-1899.
[14] L.A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior
of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl.
Math. 42 (1989), 271 - 297.
[15] C. C. Chen and C. S. Lin, Estimates of the Conformal Scalar Curvature Equation
Via the Method of Moving Planes. Comm. Pure Appl. Math. 50 (1997) 971 - 1017.
[16] V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear
elliptic PDE on Rn . Comm. Pure Appl. Math. 45 (1992), 1217 - 1269.
[17] V. Coti Zelati and P.H. Rabinowitz, Multibump periodic solutions of a family of
Hamiltonian systems. Topol. Methods Nonlinear Anal. 4 (1994), 31 - 57.
[18] M. del Pino, P. Felmer, M. Musso, Two-bubble solutions in the super-critical
Bahri-Coron’s problem. Calc. Var. Partial Differential Equations 16 (2003), 113
- 145.
[19] M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi Conjecture in Dimensions
N ≥ 9. Annals of Mathematics 174 (2011), no.3, 1485-1569.
MULTI-BUMP SOLUTIONS ON LATTICES
55
[20] M. del Pino, M. Kowalczyk and J. Wei Concentration on curves for nonlinear
Schrödinger equations. Communications on Pure and Applied Mathematics 60
(2007), no. 1, 113-146.
[21] M. del Pino, M. Kowalczyk and J. Wei, Entire solutions of Allen-Cahn equation
and complete embedded minimal surfaces with finite total curvature in R3 . J. Diff.
Geometry 83(2013), no.1, 67-131.
[22] M. del Pino, M. Kowalczyk, F. Pacard and J. Wei, The Toda system and multipleend solutions of autonomous planar elliptic problems. Advances in Mathematics
224(2010), no.4, 1462-1516.
n+2
[23] W. Ding and W.-M. Ni, On the elliptic equation ∆u + Ku n−2 = 0 and related
topics. Duke Math. J. 52 (1985), 485 - 506.
[24] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrodinger
equation with a bounded potential. J. Funct. Anal. 69 (1986), 397-408.
[25] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed
Neumann problems, J. Diff. Eqns. 158 (1999), no. 1, 1-27.
[26] N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem
with the critical exponent. Comm. Partial Differential Equations 26 (2001), 1929
- 1946.
[27] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the
maximum principle. Comm. Math. Phys. 68 (1979), 209 - 243.
[28] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of
nonlinear elliptic equations in Rn , Advances in Math. Supplementary Studies
7A(L.NACHBIN ed.) (1981) 369 - 402.
[29] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear
elliptic equations. Comm. Pure Appl. Math. 34 (1981), 525 - 598.
[30] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin 1983
[31] Z.-C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equation involving critical Sobolev exponent. Ann. Inst. Poincaré, Anal. non linéaire
8, (1991) 159 - 174.
[32] Y.Y. Li, On −∆u = K(x)u5 in R3 . Comm. Pure Appl. Math. 46 (1993), 303 340.
[33] Y.Y. Li, Prescribing scalar curvature on S3 , S4 , and related problems. J. Func.
Anal. 118 (1993), 43 - 118.
[34] Y.Y. Li, Prescribing scalar curvature on Sn and related problems, Part I. J. Diff.
Eq. 120 (1995), 319 - 410.
[35] Y.Y. Li, On a singularly perturbed elliptic equation. Advances in Differential
Equations, 2 (1997), 955 - 980.
[36] Y.Y. Li, Unpublished note, 2001.
[37] Y.Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic
equations. Comm. Partial Differential Equations 23 (1998), 487 - 545.
[38] F.H. Lin, W.M. Ni and J.C. Wei, On the number of solutions for some singularly
perturbed Neumann problems. Comm. Pure Appl. Math. 60 (2007), no. 1, 113-146.
[39] A. Malchiodi, Some new entire solutions of semilinear elliptic equations in RN .
Adv. in Math. 221 (2009), no.6, 1843-1909.
n+2
[40] W.-M. Ni, On the elliptic equation ∆u + K(x)u n−2 = 0, its generalization and
application in geometry. Indiana Univ. Math. J. 31 (1982), 493 - 529.
56
Y.Y. LI, J. WEI, AND H. XU
[41] L. Nirenberg, Topics in nonlinear functional analysis. New York Univ. Lecture
Notes, New York, 1974.
[42] M. Obata, The conjecture on conformal transformations of Riemannian manifolds. J. Diff. Geom. 6 (1971), 247 - 258.
[43] Oh, Y.-G. On positive multibump states of nonlinear Schroedinger equation under
multiple well potentials. Commun. Math. Phys. 131 (1990), 223-253.
[44] P.H. Rabinowitz,Minimax methods in critical point theory with applications to
differential equations. CBMS Regional Conference Series in Mathematics, 65.
Published for the Conference Board of the Mathematical Sciences, Washington,
DC; by the American Mathematical Society, Providence, RI, 1986.
[45] F. Pacard and M. Ritore, From the constant mean curvature hypersurfaces to the
gradient theory of phase transitions. J. Differential Geom. 64 (2003), no. 3, 359 423.
[46] O. Rey, J. Wei, Blowing up solutions for an elliptic Neumann problem with subor supercritical nonlinearity. I. N = 3. J. Funct. Anal. 212 (2004), 472 - 499.
[47] O. Rey, J. Wei, Blowing up solutions for an elliptic Neumann problem with subor supercritical nonlinearity. II. N ≥ 4. Ann. Inst. H. Poincar Anal. Non Linaire
22 (2005), 459 - 484.
[48] E. Séré, Looking for the Bernoulli shift. Ann. Inst. H. Poincaré Anal. Non Linéaire
10 (1993), 561 - 590.
[49] J. Wei and S. Yan, Infinitely many solution for prescribed curvature problem on
S N . J. Func. Anal. 258 (2010), 3048 - 3081.
[50] H. Yamabe, On a deformation of Riemannian structures on compact manifolds.
Osaka Math. J. 12 (1960) 21 - 37.
[51] S. Yan, Concentration of solutions for the scalar curvature equation on RN . J.
Diff. Eqns. 163 (2000), 239 - 264.
[52] H. Xu, Critical exponent elliptic equations: Gluing via the implicit function theorem and the moving sphere method, Thesis, 2007.
n+2
[53] H. Xu, Multi-bump solutions for −∆u = K(x)u n−2 in Rn , in preparation.
School of Mathematical Sciences, Beijing Normal University, Beijing
100875, China and Department of Mathematics, Rutgers University, NJ08854,
USA.
E-mail address: yyli@math.rutgers.edu
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2.
E-mail address: jcwei@math.ubc.ca
School of Mathematics and Statistics, Huazhong University of Science
& Technology, Wuhan 430074, P. R. China
E-mail address: hyxu@hust.edu.cn