#  Existence, Local uniqueness and periodicity of bubbling solutions for a
critical nonlinear elliptic equation

Chunhua Wang, Qingfang Wang and Jing Yang  School of Mathematics and
Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China
Normal University, Wuhan, 430079, P. R. China<EMAIL_ADDRESS>School of Mathematics and Computer Science, Wuhan Polytechnic University,
Wuhan, 430023, P. R. China<EMAIL_ADDRESS>School of Science, Jiangsu
University of Science and Technology, Zhenjiang 212003, China
<EMAIL_ADDRESS>
###### Abstract.

We revisit the following nonlinear critical elliptic equation

$-\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\;u>0\;\;\;\hbox{ in
}\mathbb{R}^{N},$

where $N\geq 5.$ Although there are some existence results of bubbling
solutions for problem above, there are no results about the periodicity of
bubbling solutions. Here we investigate some related problems. Assuming that
$Q(y)$ is periodic in $y_{1}$ with period 1 and has a local minimum at 0
satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely
many bubbling solutions of the problem above. This local uniqueness result
implies that some bubbling solutions preserve the symmetry of the potential
function $Q(y),$ i.e. the bubbling solution whose blow-up set is
$\\{(jL,0,...,0):j=0,1,2,...,m\\}$ must be periodic in $y_{1}$ provided that
$L$ is large enough, where $m$ is the number of the bubbles which is large
enough but independent of $L.$ Moreover, we also show a non-existence of this
bubbling solutions for the problem above if the local minimum of $Q(y)$ does
not equal to zero.

Key words : Multi-bubbling solutions, Critical exponents, Prescribed scalar
curvature, Local uniqueness, Periodicity.

AMS Subject Classification : 35B40, 35B45, 35J40.

##  1\.  Introduction and the main results

In this paper, we consider the following nonlinear elliptic equations with
critical exponent:

$-\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\;u>0\;\;\;\hbox{ in
}\mathbb{R}^{N},$ (1.1)

where $N\geq 5$, and $Q(y)$ is a bounded nonnegative smooth function.

This problem corresponds to the following well-known Brezis-Nireberg problem
in $S^{N},$

$\displaystyle-\Delta_{S^{N}}u=u^{\frac{N+2}{N-2}}+\mu u,\,u>0,$ (1.2)

after using the stereographic project, problem (1.2) can be reduced to (1.1)
with $Q(y)=\frac{-4\mu-N(N-2)}{(1+|y|^{2})^{2}}$ and $Q(y)>0$ if
$\mu<-\frac{N(N-2)}{4}$. Problem (1.2) has been studied widely, one can refer
to [5, 6, 13, 15].

When there is no linear term $Q(y)u$ in equation (1.1), i.e the following
well-known prescribed scalar curvature problem

$-\Delta u=K(y)u^{\frac{N+2}{N-2}},\;\;\;u>0\;\;\;\hbox{ in }\mathbb{R}^{N},$
(1.3)

where $N\geq 5$, and $K(y)$ is a bounded nonnegative smooth function. Compared
with problem (1.1), there are many results about problem (1.3). When $K(y)$ is
positive and periodic, Li proved that (1.3) has infinitely many multi-bump
solutions for $N\geq 3$ in [20, 21, 22] by gluing approximate solutions into
genuine solutions with masses concentrating near isolated sets of maximum
points of $K(y).$ Similar results were obtained by Yan in [29] provided that
$K(y)$ has a sequence of strict local maximum points tending to infinity. When
$K(y)$ is positive and periodic, in [28] Xu constructed multi-bump solutions
with mass concentrating near critical points of $K(y)$ including saddle
points. When $K(y)$ is a positive radial function with a strict local maximum
at $|y|=r_{0}>0$ and satisfies

$K(r)=K(r_{0})-|r-r_{0}|^{\alpha}+O(|r-r_{0}|^{\alpha+\iota})$

for some constant $c_{0}>0,\iota>0$ and $\alpha\in[2,N-2)$ near $|x|=r_{0}$,
Wei and Yan obtained in [27] solutions with a large number of bumps
concentrating near the sphere $|x|=r_{0}$ and $N\geq 5.$ Assuming that
$K(y)>0$ is periodic in $(x_{1},...,x_{k})$ with $1\leq k<\frac{N-2}{2},$
under some natural conditions on $K(y)$ near a critical point, Li, Wei and Xu
in [23] proved the existence of multi-bump solutions where the centers of
bumps can be placed in some lattices in $\mathbb{R}^{k},$ including infinite
lattices. They also showed that for $k\geq\frac{N-2}{2},$ no such solutions
exist. In [11], Deng, Lin and Yan obtained a local uniqueness and periodicity
result of bubbling solutions for (1.3) under the basic assumptions of [23].
For poly-harmonic prescribed scalar curvature problem, under some similar
assumptions of the prescribed scalar curvature as in [11], in [18] Guo, Peng
and Yan study the existence, local uniqueness and periodicity of a bubbling
solutions where the bubble set are a $k$-dimensional lattice. About more other
related results, one can refer to [1, 2, 3, 10, 14].

Since when $Q\geq 0$ and $Q\not\equiv 0$, the mountain pass value for problem
(1.1) is not a critical value of the corresponding functional, all the
arguments based on the concentration compactness arguments [24, 25] can not be
used to obtain an existence result of solutions for (1.1). So far, there are
very few existence results of solutions for problem (1.1). To our best
knowledge, the first existence result for problem (1.1) is from [4] where
Benci and Cerami proved that if $\|Q\|_{L^{\frac{N}{2}}(\mathbb{R}^{N})}$ is
suitably small, problem (1.1) has a solution whose energy is in the interval
$\bigl{(}\frac{1}{N}S^{\frac{N}{2}},\frac{2}{N}S^{\frac{N}{2}}\bigr{)}$, where
$S$ is the best Sobolev constant in the embedding
$D^{1,2}(\mathbb{R}^{N})\hookrightarrow L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$.
Note that the assumption $Q\in L^{\frac{N}{2}}(\mathbb{R}^{N})$ excludes the
case $Q\geq c_{0}>0$. Later, in order to cancel this restriction, in [9] Chen,
Wei and Yan has proved problem (1.1) has infinitely many non-radial solutions
where $Q(y)$ is a radial bounded positive function, and $r^{2}Q(r)$ has a
local maximum point or a local minimum point $r_{0}>0$ with $Q(r_{0})>0$.
Assuming that $Q(y)$ satisfies a weaker symmetric condition, Peng, Wang and
Yan in [26] has shown that problem (1.1) has infinitely many solutions by
combining the finite dimensional reduction argument and local Pohozaev
identities, where the concentrated points of the bubble solutions include a
saddle point of a function involving the potential function $Q(y).$ Very
recently, He, Wang and Wang in [19] have proved the non-degeneracy of the
bubble solutions constructed in [26] by some local Pohozaev identities and
also construct a new type of bubble solutions for problem (1.1).

As far as we know, the results on the uniqueness of solutions which have the
concentration phenomena are few. But it is a very profound topic to study the
uniqueness of solutions. In this aspect, the first result is the uniqueness of
solutions concentrating at one point for Dirichlet problems with critical
nonlinearity on bounded domains given by Glangetas in [16]. When the right
nonlinearity of equation (1.1) is subcritical, by calculating the number of
single-bump solutions to the equation, Grossi [17] proved that there are one
solution concentrating at any non-degenerate critical point of $Q(y)$. Later,
using topological degree, Cao and Heniz [7] gained the uniqueness of multi-
bump solutions to the subcritical problem. For more other work about the
uniqueness of solutions with the concentration phenomena, one can refer to [8,
11, 18]

In this paper, motivated by [11, 18, 26], we aim to study the existence, the
local uniqueness and the periodicity of the bubble solutions for equation
(1.1), where we mainly want to study the impact of the linear term $Q(y)u$ in
equation (1.1) to them. Here we call the solutions are local unique if two
sequences of solutions blow up at the same set. This uniqueness implies
certain kind of symmetry. We will prove that the two solutions are the same by
obtaining some useful estimates and applying some local Pohozaev identities.

We assume that $Q(y)$ satisfies the following conditions:
$(Q_{1})0\leq\inf\limits_{y\in\mathbb{R}^{N}}Q(y)<\sup\limits_{y\in\mathbb{R}^{N}}Q(y)<\infty$;
$(Q_{2})Q\in C^{1}(\mathbb{R}^{N})$ is 1-periodic in $y_{1}$;
$(Q_{3})$0 is a critical point of $Q(y)$ and there exists some real number
$\beta\in(N-4,N-2)$ such that for all $|y|$ small, it holds

$Q(y)=Q(0)+a|y|^{\beta}+O(|y|^{\beta+1})=0+a|y|^{\beta}+O(|y|^{\beta+1}),$

where $a<0.$

To state the main results of this paper, we need to introduce some notations
first. It is well known that the functions

$U_{x,\mu}=(N(N-2))^{\frac{N-2}{4}}(\frac{\mu}{1+\mu^{2}|y-x|^{2}})^{\frac{N-2}{2}},\,\mu>0,x\in\mathbb{R}^{N}$

are the only solutions to the problem

$\displaystyle-\Delta
u=u^{\frac{N+2}{N-2}},\,u>0,\,\,\hbox{in}\,\mathbb{R}^{N}.$ (1.4)

Denote $x_{j}=(jL,0,0,\cdots,0),j=0,1,2,...,m,$ where $L>0$ is a large
integer. We also denote
$W_{x_{jL},\mu_{L}}=\xi(y-x_{jL})U_{x_{jL},\mu_{L}},\,W_{\textbf{x},\mu_{L}}=\sum\limits_{j=0}^{m}W_{x_{jL},\mu_{L}}$
where $\xi\in C_{0}^{\infty}(B_{2}(0))$ satisfying $\xi(y)=\xi(|y|)$, $\xi=1$
in $B_{1}(0),$ and $\xi=0$ in $\mathbb{R}^{N}\setminus B_{2}(0).$ Applying the
cut-off function $\xi$ above can not only help us deal with the slow decay of
the bubble $U$ when the space dimension is not big but also make some
calculations much simpler.

Here we introduce the following norms which capture the decay property of the
perturbation term

$\displaystyle\|\phi\|_{*}=\sup\limits_{x\in\mathbb{R}^{N}}(\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}})|\phi(y)|,$
$\displaystyle\|f\|_{**}=\sup\limits_{x\in\mathbb{R}^{N}}(\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N+2}{2}+\tau}})|f(y)|,$

where $\tau=1+\vartheta$ and
$\vartheta\in(0,\min\\{\frac{N-4}{2},\frac{\beta}{2}\\})$ is a fixed small
constant.

The existence result in our paper is the following.

######  Theorem 1.1.

Suppose that $Q(y)\geq 0$ is bounded and belongs to $C^{1}$. If $Q(y)$
satisfies the assumptions $(Q_{1})-(Q_{3})$ and $N\geq 5$, then there is an
integer $L_{0}>0$, such that for any integer $L\geq L_{0}$ and for any
positive integer $m$ large enough which is independent of $L,$ (1.1) has a
solution $u_{L}$ of the form

$u_{L}=W_{\textbf{x},\mu_{L}}(y)+\varphi_{L}=\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}+\varphi_{L}$
(1.5)

for some $x_{jL}$ and $\mu_{L},$ with

$x_{jL}=x_{j}+o_{L}(1)\,\hbox{for\, all}\,\,j,$ (1.6)

and

$\mu_{L}=L^{\frac{N-2}{\beta-N+4}}(\bar{B}+o_{L}(1)),$ (1.7)

for some constant $\bar{B}>0$ and

$|\varphi_{L}(y)|=o_{L}(1)\sum_{j=0}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}},$
(1.8)

where $o_{L}(1)\to 0$ as $L\to+\infty.$

######  Remark 1.2.

In fact, with careful estimates of $u_{L}$ in the weighted $L^{\infty}$
spaces, similar to the proof of Theorem 2 in [23] we can prove that $u_{L}$
converges locally in $\mathbb{R}^{N}$ to a solution $u$ with the form
$u=\sum\limits_{j=0}^{\infty}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}+\varphi_{L}$ as
$m\rightarrow\infty$ independent of $L.$

Now we also give a nonexistence result of the bubbling solutions for equation
(1.1).

######  Theorem 1.3.

Assume that
$Q(y)=Q(x_{j})+a|y-x_{j}|^{\beta}+O(|y-x_{j}|^{\beta+1}),x_{j}=(jL,0,...,0),j=0,1,,,,,m$
and $Q(x_{j})\neq 0,$ where $L$ and $m$ are large integers and $m$ is
independent of $L$. Then there has no $m$-bubbles solutions of the form (1.5)
satisfying (1.6) and (1.7).

######  Remark 1.4.

From the non-existence result of Theorem 1.3, we know that the assumption
$Q(x_{j})=0,j=0,1,...,m$ is natural and necessary.

To discuss the symmetric properties of the solutions obtained in Theorem 1.1,
we proceed with the following local uniqueness result for the bubbling
solutions of (1.1).

Our main result is as follows.

######  Theorem 1.5.

Under the same assumptions as in Theorem 1.1 and $N>6,$ if $u^{(1)}_{L}$ and
$u^{(2)}_{L}$ are two sequence of solutions of problem (1.1), which satisfy
(1.5), (1.6) and (1.7), then $u^{(1)}_{L}=u^{(2)}_{L}$ provided $L>0$ is large
enough.

######  Remark 1.6.

Different from the existence result of Theorem 1.1, we assume that $N>6$ in
Theorem 1.5. This is because that the estimate of $\|\varphi_{L}\|_{*}$ is not
good enough caused by the appearance of the linear term $Q(y)u$ in euqation
(1.1). And we need to require $N>6$ in the proof of (3.15). Therefore, When
$N=5,6,$ our methods are not suitable and the local uniqueness and the
periodicity of the bubbling solutions are still open.

######  Remark 1.7.

In the proof of Theorem 1.5, in order to obtain some precise estimates
involved by the linear term $Q(y)u$ in equation (1.1), we choose the radius
$\delta$ of a small ball centered at some $x_{jL}$ satisfying
$\delta=\mu^{-\theta}_{L},$ where $\theta>0$(defined in (3.6)).

Our results here shows that concentration of the solution results in the local
uniqueness. That is, if two sequence of solutions blow up at the same set,
they must coincide.

Theorem 1.5 can be used to study some other properties of the bubbling
solutions. A direct consequence of this result is the following periodic
property of the solutions.

######  Theorem 1.8.

Under the same assumption as in Theorem 1.1 and $N>6$, and $u_{L}$ is a
solution of (1.1), which satisfies (1.5), (1.6) and (1.7), then $u_{L}$ is
$L-$periodic in $y_{1}$, provided $L>0$ is large enough.

We would like to stress that the novelty and the main difficulty in the proofs
of all the Theorems above are the techniques used to obtain some new and
delicate estimates involved by the linear term $Q(y)u$ in problem (1.1).

The structure of our paper is as follows. In Section 2, we prove the existence
and nonexistence of bubbling solutions by the finite dimensional reduction
method and a contradiction argument respectively. In Section 3, applying some
local Pohozaev identities, we prove the local uniqueness of the bubbling
solutions which can induce the periodicity. Some technical and important
estimates are put in Appendices A, B and C.

##  2\. existence and nonexistence of solutions with infinitely many bubbles

In this section, we will prove Theorems 1.1 and 1.3. To prove Theorem 1.1, we
mainly use the finite dimensional reduction method. And we use a contradiction
argument to show Theorem 1.3.

###  2.1. Linearization and the finite-dimensional problem

We first construct a bubbling solutions blowing up at finite points. We define
the function spaces $X$ and $Y$ as follows: $\phi\in X$ if
$\|\phi\|_{*}<+\infty$, while $f\in Y$if $\|f\|_{**}<+\infty$. Set

$Z_{ij}=\xi(y-x_{iL})\frac{\partial W_{x_{iL},\mu_{L}}}{\partial
x_{iL,j}},Z_{i,N+1}=\frac{\partial
W_{x_{iL},\mu_{L}}}{\partial\mu_{L}},i=0,1,\cdots,m,\,\,j=1,2,...,N$

and

$H_{m}:=\Big{\\{}\phi:\phi\in X,\int_{\mathbb{R}^{N}}\phi
W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{i,j}=0,i=0,1,\cdots,m,j=1,\cdots,N+1\Big{\\}}.$
(2.1)

Denote
$W_{\textbf{x},\mu_{L}}=\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}$,
$\varphi\in H_{m}.$ We intend to find a solution of the form
$W_{\textbf{x},\mu_{L}}+\varphi$ for problem (1.1) with $\|\varphi\|_{*}$
small enough. To achieve this goal, we first prove that for fixed
$(\textbf{x},\mu_{L})$, there exists a smooth function $\varphi\in H_{m}$ such
that

$\displaystyle-\Delta(W_{\textbf{x},\mu_{L}}+\varphi)+Q(y)(W_{\textbf{x},\mu_{L}}+\varphi)=(W_{\textbf{x},\mu_{L}}+\varphi)^{2^{*}-1}+\sum\limits_{i=0}^{m}\sum\limits_{j=1}^{N+1}c_{ij}W_{x_{jL},\mu_{L}}^{2^{*}-2}Z_{ij},$
(2.2)

for some constants $c_{ij}$. Then, we show the existence of
$(\textbf{x},\mu_{L})$ such that

$\displaystyle\int_{\mathbb{R}^{N}}(-\Delta)(W_{\textbf{x},\mu_{L}}+\varphi)Z_{ij}+Q(y)(W_{\textbf{x},\mu_{L}}+\varphi)Z_{ij}-\int_{\mathbb{R}^{N}}(W_{\textbf{x},\mu_{L}}+\varphi)^{2^{*}-1}Z_{ij}=0.$
(2.3)

With this $(\textbf{x},\mu_{L})$, it is easy to prove that all $c_{ij}$ must
be zero.

Part I: the reduction. In this part, for fixed $(\textbf{x},\mu_{L})$ we find
$\varphi(\textbf{x},\mu_{L}),$ such that $C^{1}$ in $(\textbf{x},\mu_{L})$ and
(2.4) holds. Indeed, we will apply the contraction mapping theorem to prove
the following result.

######  Proposition 2.1.

Under the assumptions of Theorem 1.1, if $L>0$ is sufficiently large, (2.2)
admits a unique solution $\varphi_{L}\in H_{m}$ such that
$\|\varphi_{L}\|_{*}\leq C(\frac{1}{\mu_{L}})^{1+\epsilon},\,|c_{l}|\leq
C(\frac{1}{\mu_{L}})^{\frac{N-2}{2}-\tau+n_{l}},$ where $\epsilon>0$ is a
small constant. Moreover, $\varphi_{L}$ is $C^{1}$ in $(\textbf{x},\mu_{L}).$

First, we consider the following linear problem

$\displaystyle-\Delta\varphi+Q(y)\varphi-(2^{*}-1)W_{\textbf{x},\mu}^{2^{*}-2}\varphi=h+\sum\limits_{i=0}^{m}\sum\limits_{j=1}^{N+1}c_{ij}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{i,j}$
(2.4) $\displaystyle\varphi\in
H_{m},\sum\limits_{j=0}^{m}\int_{\mathbb{R}^{N}}W_{x_{jL},\mu_{L}}^{2^{*}-2}Z_{j,l}\varphi=0,l=1,2,\cdots,N+1.$
(2.5)

for some real number $c_{ij}$, where $h$ is a function in $Y$.

######  Lemma 2.2.

Suppose that $\varphi_{L}$ solves (2.4) for $h=h_{L}.$ If $\|h_{L}\|_{**}$
goes to zero as $L$ goes to infinity, then so does $\|\varphi_{L}\|_{*}$.

###### Proof.

We argue by a contradiction argument. Suppose that there exist
$L\rightarrow+\infty$, and $\varphi_{L}$ solving (2.4) for $h=h_{L}$,
$\mu=\mu_{L}$ with $\|h_{L}\|_{**}\rightarrow 0$ and $\|\varphi_{L}\|_{*}\geq
C>0$. We may assume that $\|\varphi_{L}\|_{*}=1$. For simplicity, we drop the
subscript $L.$

Since $Q(y)$ is non-negative, we have

$\displaystyle|\varphi(y)|\leq$ $\displaystyle
C\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}|\varphi(z)|dz$
$\displaystyle+C\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}(|h|+|\sum\limits_{j=1}^{N+1}\sum\limits_{i=0}^{m}c_{ij}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{ij}|)dz.$

As in [26], using Lemma B.2 and B.3, we can prove

$\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}|\varphi|dz\leq
C\|\varphi\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau+\theta}},$
$\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}|h(z)|dz\leq
c\|h\|_{**}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}},$

and

$\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}|\sum\limits_{j=1}^{m}W_{x_{jL},\mu_{L}}^{2^{*}-2}Z_{j,l}|dz\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}+n_{l}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}},$

where $n_{j}=1,j=0,1,2,\cdots,N$, $n_{N+1}=-1$, $\tau,\theta$ small enough.

To estimate $c_{ij}$, $i=0,1,2,\cdots,m$, $j=1,2,\cdots,N+1$, multiplying
(2.4) by $Z_{ij}$ and integrating, we see that $c_{ij}$ satisfies

$\displaystyle\sum\limits_{j=1}^{N+1}\sum\limits_{i=0}^{m}\int_{\mathbb{R}^{N}}c_{ij}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z^{2}_{ij}=\langle-\Delta\varphi+Q(y)\varphi-$
$\displaystyle(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi,Z_{ij}\rangle-\langle
h,Z_{ij}\rangle.$ (2.6)

It follows from Lemma B.1 that

$\displaystyle|\langle h,Z_{i,j}\rangle|\leq C\mu_{L}^{n_{j}}\|h\|_{**}.$
(2.7)

By direct computation, we have

$\displaystyle|\langle Q(y)\varphi,Z_{il}\rangle|\leq$ $\displaystyle
C\|\varphi\|_{*}\int_{\mathbb{R}^{N}}\frac{\xi\mu_{L}^{\frac{N-2}{2}+n_{l}}}{(1+\mu_{L}|y-x_{iL}|)^{N-2}}\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}$
(2.8)
$\displaystyle=\frac{\mu_{L}^{n_{l}}}{\mu_{L}^{1+\epsilon}}\|\varphi\|_{*}.$
(2.9)

On the other hand, we have

$\displaystyle|\langle-\Delta\varphi-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi,Z_{i,l}\rangle|=O(\frac{\mu_{L}^{n_{l}}\|\varphi\|_{*}}{\mu_{L}^{1+\epsilon}}).$
(2.10)

Combining (2.7)-(2.10), we have

$\displaystyle\bigl{\langle}-\Delta\varphi+Q(y)\varphi-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi,Z_{i,l}\bigr{\rangle}-\bigr{\langle}h,Z_{i,l}\bigr{\rangle}=O\Big{(}\mu_{L}^{n_{l}}(\frac{\|\varphi\|_{*}}{\mu_{L}^{1+\epsilon}}+\|h\|_{*})\Big{)}.$

It is easy to check that

$\displaystyle\sum\limits_{j=1}^{m}\langle
W_{x_{jL},\mu_{L}}^{2^{*}-2}Z_{j,h},Z_{i,l}\rangle=(\bar{c}+o(1))\delta_{hl}\mu_{L}^{2n_{l}}$
(2.11)

for some constant $\bar{c}>0$.

Now inserting (2.10) and (2.11) into (2.6), we find

$\displaystyle
c_{il}=\frac{1}{\mu_{L}^{n_{l}}}(o(\|\varphi\|_{*}+o(\|h\|_{**}))),$

so

$\displaystyle\|\varphi\|_{*}\leq
c\Big{(}o(1)+\|h\|_{**}+\frac{\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau+\theta}}}{\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}}\Big{)}.$
(2.12)

We can finish the proof of this lemma by using (2.12) as in [18]. ∎

From Lemma 2.2, applying the same argument as in the proof of Proposition 4.1
in [12]. We can prove the following results.

######  Lemma 2.3.

There exist $L_{0}$ and a constant $C>0$ independent of $L$, such that for
$L>L_{0}$ and all $h\in L^{\infty}(\mathbb{R}^{N})$, problem (2.4) has a
unique solution $\varphi\equiv\mathcal{L}(h)$. Moreover

$\displaystyle\|\mathcal{L}(h)\|_{*}\leq
C\|h\|_{**},\,\,\,|c_{l}|\leq\frac{C}{\mu_{L}^{n_{l}}}\|h\|_{**}.$

Now we consider

$\displaystyle-\Delta(W_{\textbf{x},\mu_{L}}+\varphi)+Q(y)(W_{\textbf{x},\mu_{L}}+\varphi)=(W_{\textbf{x},\mu}+\varphi)^{2^{*}-1}+\sum\limits_{i=0}^{m}\sum\limits_{j=1}^{N+1}c_{ij}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{ij}$
(2.13) $\displaystyle\varphi\in
H_{m},\int_{\mathbb{R}^{N}}\sum\limits_{i=1}^{m}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{i,j}\varphi=0,j=1,2,\cdots,N+1.$

Rewrite as

$\displaystyle-\Delta\varphi+Q(y)\varphi-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi=N_{L}(\varphi)+l_{L}+\sum\limits_{i=0}^{m}\sum\limits_{j=1}^{N+1}c_{ij}W_{x_{iL},\mu_{L}}^{2^{*}-2}Z_{ij},$
(2.14)

where

$\displaystyle
N_{L}(\varphi)=(W_{\textbf{x},\mu_{L}}+\varphi)^{2^{*}-1}-W_{\textbf{x},\mu_{L}}^{2^{*}-1}-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi,$

and

$\displaystyle
l_{L}=\Big{(}W_{\textbf{x},\mu_{L}}^{2^{*}-1}-\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}^{2^{*}-1}\Big{)}-Q(y)W_{\textbf{x},\mu_{L}}+\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}\Delta\xi+2\nabla\xi\nabla(\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}).$

In order to apply the contraction mapping theorem to prove that (2.14) is
uniquely solvable, we have to estimate $N_{L}(\varphi)$ and $l_{L}$
respectively.

######  Lemma 2.4.

If $N\geq 5$, then

$\|N_{L}(\varphi)\|_{**}\leq
C\|\varphi\|_{*}^{\min\\{2^{*}-1,2\\}},\,\,\|l_{L}\|_{**}\leq\frac{C}{\mu_{L}^{\frac{N-2}{2}-\tau}}.$

###### Proof.

Since the proofs are just the same as that of Lemmas A.5 and A.6 in Appendix
A, we omit it. ∎

Proof of Proposition 2.1 First set

$\mathcal{N}=\Big{\\{}w:w\in C(\mathbb{R}^{N})\cap
H_{m},\|w\|_{*}\leq\frac{1}{\mu^{\frac{N-2}{2}-\tau-\epsilon}_{L}},\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}W_{x_{jL},\mu_{L}}^{2^{*}-2}Z_{j,l}w=0\Big{\\}},$

where $\epsilon>0$ small and $l=1,2,\cdots,N+1.$

Then (2.14) is equivalent to

$\varphi=A(\varphi)=:\mathcal{L}(N_{L}(\varphi))+\mathcal{L}(l_{L})$

where $\mathcal{L}$ is defined in Lemma 2.2. We will prove that $A$ is a
contraction map from $\mathcal{N}$ to $\mathcal{N}$. We only consider the case
$N\geq 6$ since that of $N=5$ is similar.

Firstly, we have

$\displaystyle\|A(\varphi)\|_{*}\leq
C(\|N_{L}(\varphi)\|_{**}+\|l_{L}\|_{**})\leq
C(\|\varphi\|_{*}^{\min\\{2^{*}-1,2\\}}+(\frac{1}{\mu_{L}})^{\frac{N-2}{2}-\tau})\leq\frac{C}{\mu^{\frac{N-2}{2}-\tau-\epsilon}_{L}}.$

Hence, $A$ maps $\mathcal{N}$ to $\mathcal{N}$.

On the other hand,we see

$\displaystyle\|A(\varphi_{1})-A(\varphi_{2})\|_{*}\leq
C\|N_{L}(\varphi_{1})-N_{L}(\varphi_{2})\|_{**}.$

It is easy to check that if $N\geq 6$, then

$\displaystyle|N_{L}(\varphi_{1})-N_{L}(\varphi_{2})|\leq$
$\displaystyle|N_{L}^{\prime}(\varphi_{1}+\theta\varphi_{2})||\varphi_{1}-\varphi_{2}|\leq
C(|\varphi_{1}|^{2^{*}-2}+|\varphi_{2}|^{2^{*}-2})|\varphi_{1}-\varphi_{2}|$
$\displaystyle\leq$ $\displaystyle
C(\|\varphi_{1}\|_{*}^{2^{*}-2}+\|\varphi_{2}\|_{*}^{2^{*}-2})\|\varphi_{1}-\varphi_{2}\|_{*}\Big{(}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}\Big{)}^{2^{*}-1}$
$\displaystyle\leq$ $\displaystyle
C(\|\varphi_{1}\|_{*}^{2^{*}-2}+\|\varphi_{2}\|_{*}^{2^{*}-2})\|\varphi_{1}-\varphi_{2}\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N+2}{2}+\tau}}.$

Hence

$\displaystyle\|A(\varphi_{1})-A(\varphi_{2})\|_{*}\leq
C(\|\varphi_{1}\|_{*}^{2^{*}-2}+\|\varphi_{2}\|_{*}^{2^{*}-2})\|\varphi_{1}-\varphi_{2}\|_{*}\leq\frac{1}{2}\|\varphi_{1}-\varphi_{2}\|_{*}.$

Therefore, $A$ is a contraction map.

Now by the contraction mapping theorem, there exists a unique
$\varphi\in\mathcal{N}$ such that

$\displaystyle\|\varphi\|_{*}\leq\|\mathcal{L}(N_{L}(\varphi))\|_{*}+\|\mathcal{L}(l_{L})\|_{*}\leq
C(\|N_{L}(\varphi)\|_{**}+\|l_{L}\|_{**})\leq
C(\frac{1}{\mu_{L}})^{\frac{N-2}{2}-\tau}.$

Moreover, we get the estimate of $c_{l}$ from Lemma 2.3.

Part II: the finite-dimensional problem

It follows from Lemmas A.7 and A.8, Proposition 2.1, (2.3) is equivalent to

$\displaystyle|x_{j}-x_{jL}|=O(\frac{1}{\mu_{L}^{2}}),$ (2.15)

and

$\displaystyle\sum\limits_{i\neq
j}\frac{C_{4}}{\mu_{L}^{N-2}|x_{iL}-x_{jL}|^{N-2}}-\frac{1}{\mu_{L}^{\beta+2}}=O(\frac{1}{\mu_{L}^{\beta+3}}).$
(2.16)

###  2.2. Proof of the existence result

###### Proof of Theorem 1.1.

We need to solve (2.15) and (2.16). Note that

$\displaystyle\frac{1}{|x_{i}-x_{j}|^{N-2}}=\frac{1}{|i-j|^{N-2}L^{N-2}}:=\frac{d_{ij}}{L^{N-2}},\,i\neq
j.$

So, we can use Lemma D.1 in [18] to obtain the result.

∎

###  2.3. Proof of the nonexistence result

In this subsection, we mainly use a contradiction argument to prove Theorem
1.3. To obtain a contradiction, we need to obtain some technical and careful
estimates.

###### Proof of Theorem 1.3.

For simplicity of notations, here we still denote
$W_{\textbf{x},\mu_{L}}=\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}.$
Assume that problem (1.1) has a solution $u_{\textbf{x},\mu_{L}},$ that is

$\displaystyle-\Delta\big{(}W_{\textbf{x},\mu_{L}}+\varphi_{L}\big{)}+Q(y)\big{(}W_{\textbf{x},\mu_{L}}+\varphi_{L}\big{)}=\big{(}W_{\textbf{x},\mu_{L}}+\varphi_{L}\big{)}^{2^{*}-1}.$
(2.17)

Since $U_{x_{jL},\mu_{L}}$ satisfies
$-\Delta\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}=\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}^{2^{*}-1}$,
we can rewrite (2.17) as

$\displaystyle\sum\limits_{j=0}^{m}Q(x_{j})U_{x_{jL},\mu_{L}}$ (2.18)
$\displaystyle=$
$\displaystyle-\sum\limits_{j=0}^{m}(Q(y)-Q(x_{jL}))U_{x_{jL},\mu_{L}}+\sum\limits_{j=0}^{m}(Q(x_{j})-Q(x_{jL}))U_{x_{jL},\mu_{L}}$
(2.19)
$\displaystyle+\sum\limits_{j=0}^{m}Q(y)\big{(}1-\xi(y-x_{jL})\big{)}U_{x_{jL},\mu_{L}}-\big{(}-\Delta\varphi_{L}+Q(y)\varphi_{L}-(2^{*}-1)W^{2^{*}-2}_{\textbf{x},\mu_{L}}\varphi_{L}\big{)}$
(2.20)
$\displaystyle+\Big{(}\big{(}W_{\textbf{x},\mu_{L}}+\varphi_{L}\big{)}^{2^{*}-1}-(2^{*}-1)W^{2^{*}-2}_{\textbf{x},\mu_{L}}\varphi_{L}-\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{j}}^{2^{*}-1}\Big{)}$
(2.21)
$\displaystyle+\sum\limits_{j=0}^{m}\Delta\xi(y-x_{jL})U_{x_{jL},\mu_{L}}+\sum\limits_{j=0}^{m}\nabla\xi(y-x_{jL})\cdot\nabla
U_{x_{jL},\mu_{L}}.$ (2.22)

Multiply $U_{x_{iL},\mu_{L}}$ on both sides of equation (2.18) and integrate
over $\mathbb{R}^{N}$. Applying integrating by parts, we obtain

$\displaystyle\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}Q(x_{j})U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}=-\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}(Q(y)-Q(x_{jL}))U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}$
(2.23)
$\displaystyle+\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}(Q(x_{j})-Q(x_{jL}))U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}+\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}Q(y)\big{(}1-\xi(y-x_{jL})\big{)}U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}$
(2.24)
$\displaystyle-\int_{\mathbb{R}^{N}}\big{(}-\Delta\varphi_{L}+Q(y)\varphi_{L}-(2^{*}-1)W^{2^{*}-2}_{\textbf{x},\mu_{L}}\varphi_{L}\big{)}U_{x_{iL},\mu_{L}}$
(2.25)
$\displaystyle+\int_{\mathbb{R}^{N}}\Big{(}\big{(}W_{\textbf{x},\mu_{L}}+\varphi_{L}\big{)}^{2^{*}-1}-(2^{*}-1)W^{2^{*}-2}_{\textbf{x},\mu_{L}}\varphi_{L}-\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{j}}^{2^{*}-1}\Big{)}U_{x_{iL},\mu_{L}}$
(2.26)
$\displaystyle+\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}\Delta\xi(y-x_{jL})U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}+\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}\nabla\xi(y-x_{jL})\cdot\nabla
U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}.$ (2.27)

Similar to [26], we can estimate the right hand side of (2.23) equals to
$o(\frac{1}{\mu_{L}^{2}})$ and

$\int_{\mathbb{R}^{N}}\sum\limits_{j=0}^{m}Q(x_{j})U_{x_{jL},\mu_{L}}U_{x_{iL},\mu_{L}}=\frac{1}{\mu_{L}^{2}}\int_{\mathbb{R}^{N}}Q(x_{i})U^{2}+o\big{(}\frac{1}{\mu^{2}_{L}}\big{)}.$

Observing that $\int_{\mathbb{R}^{N}}Q(x_{j})U^{2}\neq 0$ since $Q(x_{j})\neq
0,$ we get a contradiction.

∎

##  3\. local uniqueness and periodicity

###  3.1. Some estimates of the bubbling solutions

Let $u_{L}$ be a solution of (1.1), which satisfies (1.5), (1.6),(1.7). In
this section, we will estimate $\mu_{L}$ and $|x_{jL}-x_{j}|.$ The following
two Pohozaev identities play an important role in these estimates:

$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
u_{L}}{\partial\nu}\frac{\partial u_{L}}{\partial y_{i}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\nu_{i}+\frac{1}{2}\int_{\partial
B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\nu_{i}-\frac{1}{2}\int_{B_{\delta}(x_{jL})}\frac{\partial
Q}{\partial y_{i}}u_{L}^{2}$ (3.1) $\displaystyle=$
$\displaystyle\frac{1}{2^{*}}\int_{\partial
B_{\delta}(x_{jL})}u_{L}^{2^{*}}\nu_{i},$ (3.2)

and

$\displaystyle-\int\limits_{\partial B_{\delta}(x_{jL})}\frac{\partial
u_{L}}{\partial\nu}\langle y-x_{jL},\nabla
u_{L}\rangle+\frac{1}{2}\int\limits_{\partial B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\langle y-x_{jL},\nu\rangle+\frac{2-N}{2}\int\limits_{\partial
B_{\delta}(x_{jL})}\frac{\partial u_{L}}{\partial\nu}u_{L}$ (3.3)
$\displaystyle+\frac{1}{2}\int\limits_{\partial
B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\langle\nu,y-x_{jL}\rangle-\frac{1}{2}\int\limits_{B_{\delta}(x_{jL})}u_{L}^{2}\langle\nabla
Q,y-x_{jL}\rangle-\int_{B_{\delta}(x_{jL})}Q(y)u_{L}^{2}$ (3.4)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{1}{2^{*}}\int_{\partial
B_{\delta}(x_{jL})}u_{L}^{2^{*}}\langle\nu,y-x_{jL}\rangle,$ (3.5)

where $\nu$ is the outward unit normal of $\partial B_{\delta}(x_{jL})$.

We would like to point out that in order to estimate
$\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\nu_{i}$ in (3.1)
and $\frac{1}{2}\int\limits_{\partial
B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\langle\nu,y-x_{jL}\rangle$ in (3.3) more
accurately, different from [11, 18] in the sequel we take
$\delta=\mu^{-\theta}_{L}(\theta>0),$ where

$\theta=\max\Big{\\{}\frac{\beta+4-N}{\beta+1-2\tau},1\Big{\\}}.$ (3.6)

######  Lemma 3.1.

Relation (3.1) is equivalent to

$\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial Q(y)}{\partial
y_{i}}U_{x_{jL},\mu_{jL}}^{2}=O\Big{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{\beta+2}}+\max|x_{iL}-x_{i}|^{2\beta-1}\Big{)}.$
(3.7)

###### Proof.

First, we have

$\displaystyle\int_{\partial
B_{\delta}(x_{jL})}u_{L}^{2^{*}}\nu_{i}=O(\frac{1}{\mu_{L}^{N}}).$ (3.8)

and

$\displaystyle\int_{\partial
B_{\delta}(x_{jL})}Q(x)u_{L}^{2}\nu_{i}=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$
(3.9)

whose proof we put in Appendix C.

On the other hand, note that

$\displaystyle
U_{x_{jL},\mu_{L}}(y)=\frac{(N(N-2))^{\frac{N-2}{4}}}{\mu_{L}^{\frac{N-2}{2}}}\frac{1}{|y-x_{jL}|^{N-2}}+O\Big{(}\frac{1}{\mu_{L}^{\frac{N+2}{2}}}\Big{)},\,\,\,\,\,y\in\partial
B_{\delta}(x_{jL}).$

We can deduce from Corollary B.2

$\displaystyle\frac{1}{(N(N-2))^{\frac{N-2}{4}}}\Big{(}-\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial u_{L}}{\partial\nu}\frac{\partial
u_{L}}{\partial y_{i}}+\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\nu_{i}\Big{)}$ (3.10) $\displaystyle=$
$\displaystyle-\frac{1}{\mu_{jL}^{N-2}}\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial\frac{1}{|y-x_{jL}|^{N-2}}}{\partial\nu}\frac{\partial\frac{1}{|y-x_{jL}|^{N-2}}}{\partial
y_{i}}+\frac{1}{2\mu_{jL}^{N-2}}\int_{\partial
B_{\delta}(x_{jL})}|\nabla\frac{1}{|y-x_{jL}|^{N-2}}|^{2}\nu_{i}$ (3.11)
$\displaystyle+O\Big{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{N}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\frac{N-2}{2}+\tau}_{L}}+\frac{\max_{i}|x_{iL}-x_{i}|^{2\beta}}{\mu^{2\tau}_{L}}\Big{)}$
(3.12) $\displaystyle=$ $\displaystyle
O\Big{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{N}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\frac{N-2}{2}+\tau}_{L}}+\frac{\max_{i}|x_{iL}-x_{i}|^{2\beta}}{\mu^{2\tau}_{L}}\Big{)}.$
(3.13)

Moreover, from

$\displaystyle\int_{B_{\delta}(x_{jL})}U_{x_{jL},\mu_{L}}\sum\limits_{i\neq
j}U_{x_{iL},\mu_{L}}+\Big{(}\sum\limits_{i\neq
j}U_{x_{iL},\mu_{L}}\Big{)}^{2}=O\Big{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}\Big{)},$
(3.14)

and

$\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}u_{L}^{2}=\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}\Big{(}U_{x_{jL},\mu_{jL}}+\sum_{k\neq
j}\xi(y-x_{kL})U_{x_{kL},\mu_{jL}}+\varphi_{L}\Big{)}^{2}$ (3.15)
$\displaystyle=$ $\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial
Q}{\partial
y_{i}}U_{x_{jL},\mu_{jL}}^{2}+C\|\varphi_{L}\|_{*}\Big{(}\frac{1}{\mu^{\beta+1}_{L}}+|x_{jL}-x_{j}|^{\beta-1})\Big{)}+\frac{1}{\mu_{L}^{N-2}L^{N-2}}$
(3.16) $\displaystyle=$ $\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial
Q}{\partial
y_{i}}U_{x_{jL},\mu_{jL}}^{2}+C\Big{(}\frac{1}{\mu^{\beta+2}_{L}}+\max_{j}|x_{jL}-x_{j}|^{2\beta-1})+\frac{1}{\mu_{L}^{N-2}L^{N-2}}\Big{)}.$
(3.17)

Combining (3.10)-(3.15), we obtain

$\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}U_{x_{jL},\mu_{L}}^{2}=O\Big{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{\beta+2}}+\max|x_{iL}-x_{i}|^{2\beta-1}\Big{)}.$

∎

######  Lemma 3.2.

Relation (3.3) is equivalent to

$\displaystyle\frac{1}{\mu_{L}^{{\beta+2}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{i\neq
j}\frac{B}{\mu_{L}^{N-2}|x_{iL}-x_{jL}|^{N-2}}$ (3.19)
$\displaystyle+O\Big{(}\frac{1}{\mu_{L}^{N}}+\frac{\max|x_{iL}-x_{i}|}{\mu_{L}^{\beta+1}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta-1}}{\mu^{3}_{L}}+\frac{1}{\mu^{\beta+3}_{L}}\Big{)},$

where $B>0$ is a constant.

###### Proof.

Noting that for $y\in\partial B_{\delta}(x_{jL})$, we have

$\displaystyle u_{L}(y)$ $\displaystyle=$ $\displaystyle
D_{L}(y)+O\Big{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\tau}_{L}}\Big{)}$
$\displaystyle=:$
$\displaystyle(N(N-2))^{\frac{N-2}{4}}\sum\limits_{i=0}^{m}\frac{1}{\mu_{L}^{\frac{N-2}{2}}|y-x_{iL}|^{N-2}}+O\Big{(}\frac{1}{\mu_{L}^{\frac{N+2}{2}}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\tau}_{L}}\Big{)}.$

We can deduce

$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
u_{L}}{\partial\nu}\langle y-x_{jL},\nabla
u_{L}\rangle+\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\langle y-x_{jL},\nu\rangle+\frac{2-N}{2}\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial u_{L}}{\partial\nu}u_{L}$ $\displaystyle=$
$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
D_{L}}{\partial\nu}\langle y-x_{jL},\nabla
D_{L}\rangle+\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}|\nabla
D_{L}|^{2}\langle y-x_{jL},\nu\rangle$
$\displaystyle+\frac{2-N}{2}\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
D_{L}}{\partial\nu}D_{L}+O\Big{(}\frac{1}{\mu_{L}^{N}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\frac{N-2}{2}+\tau}_{L}}+\frac{\max_{i}|x_{iL}-x_{i}|^{2\beta}}{\mu^{2\tau}_{L}}\Big{)}.$

On the other hand, it is easy to check that

$\displaystyle\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial\frac{1}{|y-x_{jL}|^{N-2}}}{\partial\nu}\langle
y-x_{jL},\nabla\frac{1}{|y-x_{jL}|^{N-2}}\rangle+\frac{1}{2}\int_{\partial
B_{\delta}(x_{jL})}|\nabla\frac{1}{|y-x_{jL}|^{N-2}}|^{2}\langle
y-x_{jL},\nu\rangle$ $\displaystyle+\frac{2-N}{2}\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial\frac{1}{|y-x_{jL}|^{N-2}}}{\partial\nu}\frac{1}{|y-x_{jL}|^{N-2}}=0.$

So we find

$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
u_{L}}{\partial\nu}\langle y-x_{jL},\nabla
u_{L}\rangle+\frac{1}{2}\int_{\partial B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\langle y-x_{jL},\nu\rangle+\frac{2-N}{2}\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial u_{L}}{\partial\nu}u_{L}$ (3.20)
$\displaystyle=$ $\displaystyle B^{\prime}\sum\limits_{i\neq
j}\frac{1}{\mu_{L}^{N-2}|x_{iL}-x_{jL}|^{N-2}}+O\Big{(}\frac{1}{\mu_{L}^{N}}+\frac{\max_{i}|x_{iL}-x_{i}|^{\beta}}{\mu^{\frac{N-2}{2}+\tau}_{L}}+\frac{\max_{i}|x_{iL}-x_{i}|^{2\beta}}{\mu^{2\tau}_{L}}\Big{)},$
(3.21)

where $B^{\prime}>0$ is a constant.

Similar to (C.1), we can also obtain

$\displaystyle\int_{\partial B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\langle
y-x_{jL},\nu\rangle=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)}.$ (3.22)

On the other hand, we have

$\displaystyle\int_{B_{\delta}(x_{jL})}u_{L}^{2}\langle\nabla
Q(y),y-x_{jL}\rangle$ (3.23) $\displaystyle=$ $\displaystyle
a\int_{\mathbb{R}^{N}}\langle\nabla|y-x_{jL}|^{\beta},y-x_{jL}\rangle
u_{L}^{2}+O\Big{(}\frac{\max\limits_{i}|x_{iL}-x_{i}|}{\mu_{L}^{\beta+1}}+\frac{\max\limits_{i}|x_{iL}-x_{i}|^{\beta-1}}{\mu^{3}_{L}}+\frac{1}{\mu^{\beta+3}_{L}}\Big{)}$
(3.24) $\displaystyle=$
$\displaystyle\frac{a\beta}{\mu_{L}^{\beta+2}}\int_{\mathbb{R}^{N}}|y|^{\beta}U_{0,1}^{2}+O\Big{(}\frac{\max\limits_{i}|x_{iL}-x_{i}|}{\mu_{L}^{\beta+1}}+\frac{\max\limits_{i}|x_{iL}-x_{i}|^{\beta-1}}{\mu^{3}_{L}}+\frac{1}{\mu^{\beta+3}_{L}}\Big{)}.$
(3.25)

Since $Q(y)$ is positive and $Q(y)=a|x-x_{j}|^{\beta}+O(|x-x_{j}|^{\beta+1})$
in $B_{\delta}(x_{jL})$, similar to (3.23) we obtain

$\displaystyle\int_{B_{\delta}(x_{jL})}Q(y)u_{L}^{2}=O\Big{(}\frac{1}{\mu_{L}^{\beta+2}}\Big{)}.$
(3.26)

Finally we estimate the right hand side of (3.3)

$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}u_{L}^{2^{*}}\langle
y-x_{jL},\nu\rangle=O\Big{(}\frac{1}{\mu_{L}^{N}}\Big{)}.$ (3.27)

Thus, the result follows from (3.20)-(3.26). ∎

######  Proposition 3.3.

There holds

$|x_{jL}-x_{j}|=O(\frac{1}{\mu_{L}^{2}}).$

###### Proof.

It follows from (3.1) that

$\displaystyle\frac{1}{2}\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}u_{L}^{2}=$ $\displaystyle-\int_{\partial
B_{\delta}(x_{jL})}\frac{\partial u_{L}}{\partial\nu}\frac{\partial
u_{L}}{\partial y_{i}}+\frac{1}{2}\int_{B_{\delta}(x_{jL})}|\nabla
u_{L}|^{2}\nu_{i}$ (3.28) $\displaystyle+\frac{1}{2}\int_{\partial
B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\nu_{i}-\frac{1}{2^{*}}\int_{\partial
B_{\delta}(x_{jL})}u_{L}^{2^{*}}\nu_{i}.$ (3.29)

Similar to Lemma 3.1, we obtain

$\displaystyle-\int_{\partial B_{\delta}(x_{jL})}\frac{\partial
u_{L}}{\partial\nu}\frac{\partial u_{L}}{\partial
y_{i}}+\frac{1}{2}\int_{\Omega}|\nabla
u_{L}|^{2}\nu_{i}+\frac{1}{2}\int_{\partial
B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\nu_{i}-\frac{1}{2^{*}}\int_{\partial
B_{\delta}(x_{jL})}u_{L}^{2^{*}}\nu_{i}$ (3.30)
$\displaystyle=\frac{1}{\mu_{jL}^{N-2}|x_{iL}-x_{jL}|^{N-2}}+O\Big{(}\frac{1}{\mu_{L}^{\beta+2}}\Big{)}.$
(3.31)

On the other hand, from Lemma A.4, we have

$\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}u_{L}^{2}=$
$\displaystyle\frac{a\beta}{\mu^{2}_{L}}\int_{B_{\delta\mu_{L}}(0)}|\mu_{L}^{-1}y+x_{jL}-x_{j}|^{\beta-2}(\mu_{L}^{-1}y+x_{jL}-x_{j})U_{0,1}^{2}$
(3.32)
$\displaystyle+O\Big{(}\frac{1}{\mu^{2}_{L}}\int_{B_{\delta\mu_{L}}(0)}|\mu_{L}^{-1}y+x_{jL}-x_{j}|^{\beta}U_{0,1}^{2}\Big{)}$
(3.33) $\displaystyle=$
$\displaystyle\frac{a\beta}{\mu_{L}^{\beta+1}}\Big{(}\int_{B_{\delta\mu_{L}}(0)}|y+x_{0}|^{\beta-2}(x_{i}+x_{0,i})U_{0,1}^{2}+o(1)\Big{)}+O\Big{(}\frac{1}{\mu_{L}^{\beta+2}}\Big{)}.$
(3.34)

It follows from (3.30) and (3.32) that

$\displaystyle\int_{B_{\delta\mu_{L}}(0)}|y+x_{0}|^{\beta-2}(y_{i}+x_{0,i})U_{0,1}^{2}=o(1),$

which yields $x_{0}=0.$

Observing that
$\int_{B_{\delta\mu_{L}}(0)}|y_{i}|^{\beta-2}y_{i}U_{0,1}^{2}=0$, we get

$\displaystyle\int_{B_{\delta}(x_{jL})}\frac{\partial Q}{\partial
y_{i}}u_{L}^{2}=\frac{a\beta}{\mu_{L}^{\beta+1}}\int_{B_{\delta\mu_{L}}(0)}|y|^{\beta-2}U_{0,1}^{2^{*}}\mu_{L}(x_{jL}-x_{j})_{i}+O\Big{(}\frac{1}{\mu_{L}^{\beta+2}}\Big{)}.$

So we find

$\displaystyle\mu_{L}(x_{jL}-x_{j})_{i}=O\Big{(}\frac{1}{\mu_{L}}\Big{)},$

which shows

$\displaystyle|x_{jL}-x_{j}|=O\Big{(}\frac{1}{\mu_{L}^{2}}\Big{)}.$

∎

######  Remark 3.4.

Applying Lemma 3.2 and Proposition 3.3, we can prove that to obtain a bubbling
solution such that the center of different bubbles are separated from each
other, the constant $\beta$ in assumption $(Q_{3})$ must satisfy $\beta>N-4.$
In fact, if $\beta\leq N-4,$ then (3.19) yields

$\frac{1}{\mu^{\beta+2}_{L}}=O\Big{(}\frac{1}{\mu^{N-2}_{L}L^{N-2}}+\frac{1}{\mu^{\beta+3}_{L}}\Big{)},$

which is impossible if $\beta\leq N-4.$

Now, we prove the following Proposition.

######  Proposition 3.5.

It holds

$\displaystyle\mu_{L}=L^{\frac{N-2}{\beta-N+4}}\Big{(}\bar{B}+\frac{1}{\mu_{L}}\Big{)},$

for some constant $\bar{B}>0$.

###### Proof.

By Proposition 3.3 $|x_{j}-x_{jL}|=O(\frac{1}{\mu_{L}^{2}})$, we find

$\displaystyle\frac{1}{|x_{iL}-x_{jL}|^{N-2}}=\frac{1}{(|x_{i}-x_{j}|+O(\frac{1}{\mu_{L}^{2}}))^{N-2}}=\frac{1}{|x_{i}-x_{j}|^{N-2}}\Big{(}1+\frac{1}{\mu_{L}^{2}}\Big{)}.$

As a result, we see that (3.19) is equivalent to

$\displaystyle\frac{1}{\mu_{L}^{\beta+2}}=B\sum\limits_{i\neq
j}\frac{1}{\mu_{L}^{N-2}|x_{i}-x_{j}|^{N-2}}+O\Big{(}\frac{1}{\mu_{L}^{\beta+3}}+\frac{1}{\mu_{L}^{N}}\Big{)}.$
(3.35)

We can easily deduce from (3.35) that

$\displaystyle c_{0}L^{\frac{N-2}{\beta-N+4}}\leq\mu_{L}\leq
c_{1}L^{\frac{N-2}{\beta-N+4}}.$

Let
$\frac{1}{\mu_{L}^{\frac{N-2}{2}}}=\frac{a_{iL}}{L^{\frac{(N-2)^{2}}{2(\beta-N+4)}}}$.
Then $0<c_{0}\leq a_{jL}\leq c_{1}<+\infty$, and from [18] Lemma C.1, we
obtain

$\displaystyle a_{jL}^{\beta}=\bar{B}\sum\limits_{i\neq
j}\frac{a_{iL}}{|i-j|^{N-2}}+O(\frac{1}{\mu_{L}}).$

Thus, we complete the result. ∎

###  3.2. Local uniqueness

Assume that problem (1.1) has two different solutions $u^{(1)}_{L}$ and
$u^{(2)}_{L}$, which blow up at $x_{j}$, $j=0,1,2,\cdots,m$. For $l=1,2$, we
will use $x_{jL}^{(l)}$ and $\mu_{L}^{(l)}$ to denote the center and the
height of the bubbles appearing in $u^{(l)}_{L},$ respectively.

Let

$\eta_{L}=\frac{u^{(1)}_{L}-u^{(2)}_{L}}{\|u^{(1)}_{L}-u^{(2)}_{L}\|_{*}}.$
(3.36)

Then, $\eta_{L}$ satisfies $\|\eta_{L}\|_{*}=1$ and

$-\Delta\eta_{L}+Q(y)\eta_{L}=f(y,u^{(1)}_{L},u^{(2)}_{L}),$ (3.37)

where

$f(y,u^{(1)}_{L},u^{(2)}_{L})=\frac{1}{\|u^{(1)}_{L}-u^{(2)}_{L}\|_{*}}\bigl{(}(u^{(1)}_{L})^{2^{*}-1}-(u^{(2)}_{L})^{2^{*}-1}\bigr{)}.$
(3.38)

Write

$f(y,u^{(1)}_{L},u^{(2)}_{L})=c_{L}(y)\eta_{L}(y),$ (3.39)

where

$c_{L}(y)=(2^{*}-1)\int_{0}^{1}\bigl{(}tu^{(1)}_{L}(y)+(1-t)u^{(2)}_{L}(y)\bigr{)}^{2^{*}-2}\,dt.$
(3.40)

It follows from Propositions 3.3 and 3.5 that

$\begin{split}U_{x^{(1)}_{iL},\mu^{(1)}_{L}}-U_{x^{(2)}_{iL},\mu^{(2)}_{L}}=&O\Bigl{(}|x^{(1)}_{iL}-x^{(2)}_{iL}||\nabla
U_{x^{(1)}_{iL},\mu^{(1)}_{L}}|+|\mu^{(1)}_{L}-\mu^{(2)}_{L}||\partial_{\mu}U_{x^{(1)}_{iL},\mu^{(1)}_{L}}|\Bigr{)}\\\
=&O\Bigl{(}\frac{1}{\mu_{L}}U_{x^{(1)}_{iL},\mu^{(1)}_{L}}\Bigr{)},\end{split}$

which gives

$u_{L}^{(1)}-u_{L}^{(2)}=O\Bigl{(}\frac{1}{\mu_{L}}\sum_{i=0}^{m}U_{x^{(1)}_{iL},\mu^{(1)}_{L}}+|\varphi_{L}^{(1)}|+|\varphi_{L}^{(2)}|\Bigr{)}.$
(3.41)

Thus, we have proved

$\begin{split}c_{L}(y)=&(2^{*}-1)U^{2^{*}-2}_{x^{(1)}_{jL},\mu^{(1)}_{L}}+O\Bigl{(}\bigl{(}\frac{1}{\mu_{L}}U_{x^{(1)}_{jL},\mu^{(1)}_{L}}\cr&+\frac{1}{\mu_{L}^{\frac{N-2}{2}}L^{N-2}}+|\varphi_{L}^{(1)}|+|\varphi_{L}^{(2)}|\bigr{)}^{2^{*}-2}\Bigr{)},\quad
y\in B_{\delta}(x^{(1)}_{jL}).\end{split}$ (3.42)

It is not difficult to deduce from Lemma A.3 that

$\displaystyle|c_{L}(y)|\leq
C(W^{2^{*}-2}_{\textbf{x}^{(1)},\mu^{(1)}_{L}}(y)+W_{\textbf{x}^{(2)},\mu^{(2)}_{L}}^{2^{*}-2}(y)).$

Hence Lemma A.4 implies that there exists a large $R>0$ such that

$\displaystyle|\eta_{L}(y)|=o(1),\,\,\,\,\,\,y\in\mathbb{R}^{N}\setminus\cup_{j}B_{R(\mu^{(1)}_{L})^{-1}}(x_{jL}^{(1)}).$

To obtain a contradiction, we only need to check that $|\eta_{L}(y)|=o(1)$ in
$\cup_{j}B_{R(\mu^{(1)}_{L})^{-1}}(x^{(1)}_{jL})$, which will be achieved by
using the Pohozaev identities in the small ball $B_{\delta}(x^{(1)}_{jL}).$

Let

$\tilde{\eta}_{Lj}(y)=\bigl{(}\frac{1}{\mu^{(1)}_{L}}\bigr{)}^{\frac{N-2}{2}}\eta_{L}(\frac{1}{\mu^{(1)}_{L}}y+x^{(1)}_{jL}).$
(3.43)

######  Lemma 3.6.

It holds

$\tilde{\eta}_{Lj}(y)\to\sum_{l=0}^{N}b_{jl}\psi_{l}(y),\hbox{ as
}L\to\infty,$ (3.44)

uniformly in $C^{1}(B_{R}(0))$ for any $R>0$, where $b_{jl}$, $l=0,\cdots,N$,
are some constants, and

$\psi_{0}=\frac{\partial
U_{0,\mu_{L}}}{\partial\mu_{L}}\Bigr{|}_{\mu_{L}=1},\quad\psi_{j}=\frac{\partial
U_{0,1}}{\partial y_{j}},\;\;j=1,\cdots,N.$ (3.45)

###### Proof.

In view of $|\tilde{\eta}_{L,j}|\leq C$ in any compact subset of
$\mathbb{R}^{N}$, we may assume that $\tilde{\eta}_{L,j}\to\eta_{j}$ in
$C_{loc}(\mathbb{R}^{N})$. Then it follows from the elliptic regularity theory
and (3.37) and (3.42) that $\eta_{j}$ satisfies

$-\Delta\eta_{j}=(2^{*}-1)U_{0,1}^{2^{*}-2}\eta_{j},\quad\text{in}\;\mathbb{R}^{N},$
(3.46)

which combining with the non-degeneracy of $U_{0,1}$ gives
$\eta_{j}=\sum\limits_{l=0}^{N}b_{jl}\psi_{l}$. ∎

Let $G(y,x)=C_{N}|y-x|^{2-N}$ be the corresponding Green’s function of
$-\Delta$ in $\mathbb{R}^{N},$ where $C_{N}=(N(N-2)\omega_{N-1})^{-1}.$

######  Lemma 3.7.

There holds

$\begin{split}|\eta_{L}(x)|\leq&\sum_{j=0}^{m}\sum_{|\alpha|=0}^{1}A_{j,L,\alpha}\partial^{\alpha}G(x^{(1)}_{jL},x)+O\bigl{(}\frac{1}{\mu_{L}^{\frac{N+2}{2}-\theta}}\bigr{)}\\\
:=&\sum_{j=0}^{m}F_{jL}(x)+O\bigl{(}\frac{1}{\mu_{L}^{\frac{N+2}{2}-\theta}}\bigr{)},\quad\text{\hbox{in}
$C^{1}\bigl{(}\mathbb{R}^{N}\setminus\cup_{j=0}^{m}B_{2\delta}(x^{(1)}_{jL})\bigr{)}$},\end{split}$
(3.47)

where $\delta>0$ is any small constant, and the constants $A_{j,L,\alpha}$
satisfy the following estimates:

$A_{j,L,0}=\int_{B_{\sigma}(x^{(1)}_{j,L})}f(y,u^{(1)}_{L}(y),u^{(2)}_{L}(y))\,dy=o\bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}}}\bigr{)},$
(3.48)
$A_{j,L,\alpha}=O\bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}+|\alpha|}}\bigr{)},\quad|\alpha|\geq
1.$ (3.49)

###### Proof.

Denote $f_{L}^{*}(y)=f(y,u^{(1)}_{L}(y),u^{(2)}_{L}(y))$. We have

$\displaystyle|\eta_{L}(x)|=\int_{\mathbb{R}^{N}}G(y,x)f_{L}^{*}(y)\,dy-\int_{\mathbb{R}^{N}}G(y,x)Q(y)|\eta_{L}(y)|dy$
(3.50)

Note that

$\begin{split}\int_{\mathbb{R}^{N}}G(y,x)f_{L}^{*}(y)=&\int_{\bigcup\limits_{j=0}^{m}B_{\delta}(x^{(1)}_{jL})}G(y,x)f_{L}^{*}(y)\,dy+\int_{\mathbb{R}^{N}\setminus\cup_{j}B_{\delta}(x^{(1)}_{jL})}G(y,x)f_{L}^{*}(y)\,dy\\\
=&\int_{\bigcup\limits_{j=0}^{m}B_{\delta}(x^{(1)}_{jL})}G(x^{(1)}_{jL},x)f_{L}^{*}(y)+\sum\limits_{j=0}^{m}\int_{B_{\delta}(x^{(1)}_{jL})}G(y,x)-G(x_{jL}^{1},x)f_{L}^{*}(y)\\\
&+\int_{\mathbb{R}^{N}\setminus\cup_{j}B_{\delta}(x^{(1)}_{jL})}G(y,x)f_{L}^{*}(y)\,dy\\\
=&\sum\limits_{j=0}^{m}\sum\limits_{|\alpha|=0}^{1}A_{j,L,\alpha}\partial^{\alpha}G(x^{(1)}_{j,L},x)+\sum\limits_{j=0}^{m}O\Bigl{(}\int_{B_{\delta}(x^{(1)}_{jL})}|y-x^{(1)}_{jL}|^{2}|f^{*}_{L}(y)|\,dy\Bigr{)}\\\
&+\int_{\mathbb{R}^{N}\setminus\cup_{j}B_{\delta}(x^{(1)}_{jL})}G(y,x)f_{L}^{*}(y)\,dy.\end{split}$
(3.51)

And

$\displaystyle\int_{\mathbb{R}^{N}}G(y,x)Q(y)|\eta_{L}(x)|dy$ (3.52)
$\displaystyle=\int_{\cup
B_{\delta}(x_{jL}^{(1)})}G(y,x)|y-x_{jL}|^{\beta}|\eta_{L}|dy+\int_{\mathbb{R}^{N}\backslash\cup
B_{\delta}(x_{jL}^{(1)})}G(y,x)|y-x_{jL}|^{\beta}|\eta_{L}|dy$ (3.53)
$\displaystyle=O\Big{(}\frac{1}{\mu_{L}^{\beta+2}}\Big{)}$ (3.54)

For $y\in\mathbb{R}^{N}\setminus\cup_{j}B_{\delta}(x^{(1)}_{jL})$, noting that
$\tau=\frac{N-2}{2}-\vartheta$ for $\vartheta>0$ small, similar to [18], we
find

$\begin{split}|f_{L}^{*}(y)|\leq&C\mu_{L}^{-\tau}\bigl{(}\frac{1}{\mu_{L}^{2}}+(\mu_{L}^{-\tau}\|\varphi^{(1)}_{L}\|_{*})^{2^{*}-2}\bigr{)}\Bigl{(}\sum_{j=0}^{m}\frac{1}{|y-x^{(1)}_{jL}|^{\frac{N-2}{2}+\tau}}\Bigr{)}^{2^{*}-1}\\\
\leq&C\mu_{L}^{-\frac{N+2}{2}+\vartheta}\sum_{j=0}^{m}\frac{1}{|y-x^{(1)}_{jL}|^{\frac{N+2}{2}+\tau}}.\end{split}$

Thus, we have

$\int_{\mathbb{R}^{N}\setminus\cup_{j}B_{\delta}(x^{(1)}_{jL})}G(y,x)f_{L}^{*}(y)\,dy\leq
C\mu_{L}^{-\frac{N+2}{2}+\vartheta}\sum_{j=0}^{m}\frac{1}{|x-x^{(1)}_{jL}|^{\frac{N-2}{2}+\tau}}\leq\frac{C}{\mu_{L}^{\frac{N+2}{2}-\vartheta}}.$
(3.55)

Similarly, by Lemma A.3, we obtain

$\begin{split}&\int_{B_{\delta}(x^{(1)}_{jL})}|y-x_{jL}^{(1)}|^{2}|f^{*}_{L}|dy\\\
\leq&C\int_{B_{\delta}(x^{(1)}_{jL})}|y-x_{jL}^{(1)}|^{2}\frac{(\mu^{(1)}_{L})^{\frac{N+2}{2}}}{(1+\mu^{(1)}_{jL}|y-x^{(1)}_{jL}|)^{(\frac{N-2}{2}+\tau)(2^{*}-1)}}\\\
\leq&\frac{C}{(\mu^{(1)}_{L})^{\frac{N+2}{2}-(2^{*}-1)\vartheta}}\int_{B_{\delta}(x^{(1)}_{jL})}\frac{1}{|y-x^{(1)}_{jL}|^{N-\vartheta(2^{*}-1)}}\leq\frac{C}{\mu_{L}^{\frac{N+2}{2}-(2^{*}-1)\vartheta}}.\end{split}$
(3.56)

Inserting (3.51)-(3.56) into (3.50), we obtain (3.47). Similarly, we can prove
that (3.47) holds in
$C^{1}\bigl{(}\mathbb{R}^{N}\setminus\cup_{j=0}^{m}B_{2\delta}(x^{(1)}_{jL})\bigr{)}$.
It remains to estimate $A_{j,L,\alpha}$.

$\begin{split}A_{j,L,0}=&\int_{B_{\delta}(x^{(1)}_{jL})}f(y,u^{(1)}_{L}(y),u^{(2)}_{L}(y))\,dy\\\
=&\frac{1}{(\mu_{L}^{(1)})^{N-2}}\int_{B_{R}(0)}\frac{1}{(\mu_{jL}^{(1)})^{2}}f^{*}_{L}(\frac{1}{\mu^{(1)}_{L}}y+x^{(1)}_{jL})\,dy\\\
&+\frac{1}{(\mu_{L}^{(1)})^{\frac{N-2}{2}}}O\Bigl{(}\int_{B_{\delta\mu_{L}^{(1)}}(0)\setminus
B_{R}(0)}\frac{1}{|y|^{(N-2-\vartheta)(2^{*}-1)}}\,dy\Bigr{)}\\\
=&\frac{1}{(\mu_{L}^{(1)})^{\frac{N-2}{2}}}\Bigl{(}2^{*}-1)\int_{\mathbb{R}^{N}}U_{0,1}^{2^{*}-2}\sum_{l=0}^{N}b_{jl}\psi_{l}+o(1)\Bigr{)}=o\bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}}}\bigr{)}.\end{split}$
(3.57)

If $|\alpha|\geq 1$, then

$|A_{j,L,\alpha}|\leq
C\int_{B_{\delta}(x^{(1)}_{jL})}|y-x^{(1)}_{jL}|^{|\alpha|}|f(y,u^{(1)}_{L}(y),u^{(2)}_{L}(y))|=O\bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}+|\alpha|}}\bigr{)}.$
(3.58)

Thus, we complete the proof. ∎

Using (3.1) and (3.3), we can deduce the following identities:

$\begin{split}&-\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\frac{\partial\eta_{L}}{\partial\nu}\frac{\partial
u_{L}^{(1)}}{\partial y_{i}}-\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\frac{\partial\eta_{L}}{\partial y_{i}}\frac{\partial
u_{L}^{(1)}}{\partial\nu}+\frac{1}{2}\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\langle\nabla(u_{L}^{(1)}+u_{L}^{(2)}),\nabla\eta_{L}\rangle\nu_{i}\\\
=&\int_{\partial
B_{\delta}(x^{(1)}_{jL})}C_{L}(y)\eta_{L}\nu_{i}+\int_{\partial
B_{\delta}(x^{(1)}_{jL})}Q(x)(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L}\nu_{i}-\int_{B_{\delta}(x^{(1)}_{jL})}\frac{\partial
Q}{\partial y_{i}}(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L},\end{split}$ (3.59)

and

$\begin{split}&-\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\frac{\partial\eta_{L}}{\partial\nu}\langle
y-x_{jL}^{(1)},\nabla u_{L}^{(1)}\rangle-\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\frac{\partial u_{L}^{(2)}}{\partial\nu}\langle
y-x_{jL}^{(1)},\nabla\eta_{L}\rangle\\\ &+\frac{1}{2}\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\langle\nabla(u_{L}^{(1)}+u_{L}^{(2)}),\nabla\eta_{L}\rangle\langle
y-x_{jL}^{(1)},\nu\rangle+\frac{2-N}{2}\int_{\partial
B_{\delta}(x^{(1)}_{jL})}\frac{\partial\eta_{L}}{\partial\nu}u_{L}^{(1)}\\\
&+\frac{2-N}{2}\int_{\partial B_{\delta}(x^{(1)}_{jL})}\frac{\partial
u_{L}^{(2)}}{\partial\nu}\eta_{L}+\int_{\partial
B_{\delta}(x^{(1)}_{jL})}Q(y)(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L}\langle
y-x_{jL},\nu\rangle\\\
&=\int_{B_{\delta}(x^{(1)}_{jL})}Q(y)(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L}+\int_{B_{\delta}(x^{(1)}_{jL})}(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L}\langle\nabla
Q,x-x_{jL}^{(1)}\rangle\\\ &+\int_{\partial
B_{\delta}(x^{(1)}_{jL})}C_{L}(y)\eta_{L}\bigl{\langle}y-x^{(1)}_{jL},\nu\bigr{\rangle},\end{split}$
(3.60)

where $C_{L}(y)=\int_{0}^{1}(tu^{(1)}_{L}+(1-t)u^{(2)}_{L})^{2^{*}-1}\,dt$ and
$d>0$ is a small constant.

Similar to (3.42), we can deduce

$C_{L}(y)=U^{2^{*}-1}_{x^{(1)}_{jL},\mu^{(1)}_{L}}+O\Bigl{(}\bigl{(}\frac{1}{\mu_{L}}U_{x^{(1)}_{jL},\mu^{(1)}_{L}}+\frac{1}{\mu_{L}^{\frac{N-2}{2}}L^{N-2}}+|\varphi_{L}^{(1)}|+|\varphi_{L}^{(2)}|\bigr{)}^{2^{*}-1}\Bigr{)},\;\;y\in
B_{\delta}(x^{(1)}_{jL}).$ (3.61)

To estimate the boundary terms in (3.59) and (3.60), we need the following
estimates which can be deduced from Proposition B.1 and Lemma 3.7.

$\displaystyle
u^{(l)}_{L}(y)=\sum_{j=0}^{m}\frac{(N(N-2))^{\frac{N-2}{4}}}{\mu_{L}^{\frac{N-2}{2}}}\frac{1}{|y-x_{jL}|^{N-2}}+O(\frac{1}{\mu_{L}^{\frac{N+2}{2}}}),\,\,\,\,\,y\in\partial
B_{\delta}(x_{jL}),\,\,l=1,2,$ (3.62)

and

$\displaystyle\eta_{L}(y)=F_{jL}(y)+o(\frac{1}{\mu_{L}^{\frac{N-2}{2}}L^{N-2}})\,\,\,\,\,y\in\partial
B_{\delta}(x_{jL}).$

###### Proof of Theorem 1.5 .

Step 1. We prove $b_{j,L}=0$, $j=1,\cdots,N$. We need to estimate each term in
(3.59). From (3.61), we obtain

$\int_{\partial
B_{\delta}(x^{(1)}_{jL})}C_{L}(y)\eta_{L}\nu_{i}=O\bigl{(}\frac{1}{\mu_{L}^{N}}b_{ji}\int_{\mathbb{R}^{N}}U_{0,1}\psi_{i}+o(1)\bigr{)},$
(3.63)

and

$\begin{split}\int_{B_{\delta}(x^{(1)}_{jL})}\frac{\partial Q(y)}{\partial
y_{i}}D_{L}(y)\eta_{L}=&\frac{\beta}{\mu_{L}^{\beta-1+\frac{N+2}{2}}}\int_{B_{\delta\mu_{L}}(0)}|y|^{\beta-2}y_{i}D_{L}(\frac{y}{\mu_{jL}}+x_{jL}^{1})\tilde{\eta}_{L}\cr=&\frac{\beta}{\mu^{\beta-1+2}}\int_{B_{R}(0)}|y_{i}|^{\beta-2}y_{i}U_{0,1}\sum\limits_{k=0}^{N}b_{jL}\psi_{k}+O(\frac{1}{\mu_{L}^{\beta+2}})\cr=&\frac{1}{\mu_{L}^{\beta+1}}b_{ji}\int_{\mathbb{R}^{N}}|y_{i}|^{\beta-2}y_{i}U_{0,1}\psi_{i}+o(1)+O(\frac{1}{\mu_{L}^{\beta+2}}),\end{split}$
(3.64)

and

$\displaystyle\int_{\partial
B_{\delta}(x^{(1)}_{jL})}Q(y)(u_{L}^{(1)}+u_{L}^{(2)})\eta_{L}\nu_{i}=O(\frac{1}{\mu_{L}^{\beta+2}}).$
(3.65)

Combining (3.63), (3.64) and (3.65), we are led to

$\begin{split}\text{RHS of
\eqref{14-11-9}}=\frac{1}{\mu_{L}^{\beta+1}}b_{ji}\int_{\mathbb{R}^{N}}|y_{i}|^{\beta-2}y_{i}U_{0,1}\psi_{i}.\end{split}$
(3.66)

To estimate the left hand side of (3.59), using (3.62), we have

$\text{LHS of
\eqref{14-11-9}}=o(\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{N-1}}).$
(3.67)

So (3.66) and (3.67) yield

$\begin{split}b_{j,i}\int_{\mathbb{R}^{N}}|y|^{\beta-2}y_{i}U_{0,1}\psi_{i}+o(1)=&(\mu^{(1)}_{L})^{\beta+1}o(\frac{1}{\mu_{L}^{N-2}L^{N-2}}+\frac{1}{\mu_{L}^{N-1}})=o(1),\end{split}$
(3.68)

which implies $b_{j,i}=0,i=1,\cdots,N$.

Step 2. We prove $b_{j,0}=0$. It follows from Lemma 2.6 and (3.62) that

$\begin{split}&\text{LHS of \eqref{15-11-9}}\\\ =&\int_{\partial
B_{\delta}(x_{jL}^{(1)})}\frac{\partial\eta_{L}^{*}}{\partial\nu}\langle
x-x_{jL}^{(1)},\nabla\frac{1}{|x-x_{jL}^{(1)}|^{N-2}}\rangle+\int_{\partial
B_{\delta}(x_{jL}^{(1)})}\frac{\partial\frac{1}{|x-x_{jL}^{(1)}|^{N-2}}}{\partial\nu}\langle
x-x_{jL}^{(1)},\nabla\eta_{Lj}^{*}\rangle\\\ &+\frac{1}{2}\int_{\partial
B_{\delta}(x_{jL}^{(1)})}\langle\nabla\frac{1}{|x-x_{jL}^{(1)}|^{N-2}},\nabla\eta_{Lj}^{*}\rangle\langle
x-x_{jL}^{(1)},\nu\rangle\\\ &+\frac{2-N}{2}\int_{\partial
B_{\delta}(x_{jL}^{(1)})}\frac{\partial\eta_{Lj}^{*}}{\partial\nu}\frac{1}{|x-x_{jL}^{(1)}|^{N-2}}+\int_{\partial
B_{\delta}(x_{jL}^{(1)})}\frac{\partial\frac{1}{|x-x_{jL}^{(1)}|^{N-2}}}{\partial\nu}\eta_{Lj}^{*})\\\
&+o\bigl{(}\frac{1}{\mu_{L}^{N-2}|x_{iL}-x_{jL}|^{N-2}}\bigr{)}+O\bigl{(}\frac{1}{\mu_{L}^{N}}\bigr{)}.\end{split}$
(3.69)

If we replace $\eta_{jL}^{*}$ by $\frac{A_{jL}}{|x-x_{jL}|^{N-2}}$, then it is
easy to check that the sum of all the integrals is zero.

Therefore

$\text{LHS of
\eqref{15-11-9}}=o\bigl{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}\bigr{)}+O\bigl{(}\frac{1}{\mu_{L}^{N}}\bigr{)}.$
(3.70)

Combining (3.66) and (3.70), we are led to

$b_{j,0}\int_{\mathbb{R}^{N}}|y|^{\beta}U_{0,1}^{2^{*}-1}\psi_{0}=o(1).$
(3.71)

This gives $b_{j,0}=0$. ∎

###### Proof of Theorem 1.8.

To prove that $u_{L}$ is periodic in $y_{1}$, we let

${v_{L}(y)}=u_{L}(y_{1}-L,y_{2},\cdots,y_{N}).$

Then, $v_{L}$ is a bubbling solution whose blow-up set is the same as that of
$u_{L}$. By the local uniqueness, $v_{L}=u_{L}$. ∎

##  Appendix A Some basic estimates

In this section, we give some technical estimates. Throughout Appendices A, B,
C, we will use the same notations as before and we also use the same $C$ to
denote different constants unless otherwise stated.

First we give some known estimates which can be found in [27, 18].

######  Lemma A.1.

(Lemma B.1, [27]) Let $x_{i},x_{j}\in\mathbb{R}^{N},x_{i}\not=x_{j}$, $i\neq
j.$ It holds

$\begin{split}&\frac{1}{(1+|y-x_{i}|)^{\alpha}(1+|y-x_{j}|)^{\beta}}\\\
&\leq\frac{C}{(1+|x_{i}-x_{j}|)^{\sigma}}\Big{(}\frac{1}{(1+|y-x_{j}|)^{\alpha+\beta-\sigma}}+\frac{1}{(1+|y-x_{i}|)^{\alpha+\beta-\sigma}}\Big{)},\end{split}$

where $\alpha$ and $\beta$ are some positive constants,
$0<\sigma\leq\min(\alpha,\beta)$.

######  Lemma A.2.

(Lemma B.1, [27]) For any constant $0<\sigma<N-2$, there exists a consta

$\int_{\mathbb{R}^{N}}\frac{dz}{|y-z|^{N-2}(1+|z|)^{2+\sigma}}\leq\frac{C}{(1+|y|)^{\sigma}}.$

######  Lemma A.3.

(Lemma A.3, [18]) For any $\gamma>1,$ there exists a constant $C$, such that

$\sum_{j}\frac{1}{(1+\mu_{L}|y-x_{j}|)^{\gamma}}\leq\frac{C}{(1+\mu_{L}|y-x_{i}|)^{\theta}},\quad
y\in B_{i}:=B_{1}(x_{i}).$ (A.1)

Let $u_{L}$ be a solution of (1.1) with the form

$u_{L}=W_{\textbf{x},\mu_{L}}+\varphi_{L},\quad
W_{\textbf{x},\mu_{L}}=\sum_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}=:\sum\limits_{j=0}^{m}W_{x_{jL},\mu_{L}},$
(A.2)

satisfying (1.6), (1.7) and (1.8). It is easy to see that $\varphi_{L}$
satisfies the following equation:

$-\Delta\varphi_{L}-Q(y)\varphi_{L}-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}=N_{L}(\varphi_{L})+l_{L},$
(A.3)

where

$\displaystyle l_{L}=$
$\displaystyle(W_{\textbf{x},\mu_{L}})^{2^{*}-1}-\sum\limits_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}^{2^{*}-1}-Q(y)W_{\textbf{x},\mu_{L}}$
(A.4)
$\displaystyle+\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}\Delta\xi+2\nabla\xi\nabla\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}},$
(A.5)

and

$N_{L}(\varphi_{L})=(W_{\textbf{x},\mu_{L}}+\varphi_{L})^{2^{*}-1}-W_{\textbf{x},\mu_{L}}^{2^{*}-1}-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}.$
(A.6)

By assumption (1.8), we have $\|\varphi_{L}\|_{*}\to 0$ as $L\to+\infty$.

######  Lemma A.4.

It holds

$u_{L}\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}},$
(A.7)

and

$\varphi_{L}\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}.$
(A.8)

###### Proof.

Since the proof is the same as that of Lemma A.3 in [11], here we omit it. ∎

We now estimate $N_{L}(\varphi_{L})$ and $l_{L}$.

######  Lemma A.5.

If $N\geq 5$, then

$\|N_{L}(\varphi_{L})\|_{**}\leq C\|\varphi_{L}\|_{*}^{\min(2^{*}-1,2)}.$

###### Proof.

Considering that the proof is just the same as that of Lemma 2.4 in [26], we
omit it. ∎

######  Lemma A.6.

If $N\geq 5$, then

$\|l_{L}\|_{**}\leq\frac{C}{\mu_{L}^{\frac{N-2}{2}-\tau}}+C\max_{j}{|x_{jL}-x_{j}|^{\beta}}.$

###### Proof.

Recall that from (A.4)

$\displaystyle l_{L}$
$\displaystyle=\Big{[}(W_{\textbf{x},\mu_{L}})^{2^{*}-1}-\sum\limits_{j=0}^{m}\xi
U_{x_{jL},\mu_{L}}^{2^{*}-1}\Big{]}-Q(y)W_{\textbf{x},\mu_{L}}$
$\displaystyle\quad+\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}\Delta\xi(y-x_{jL})+2\nabla\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}\nabla\xi(y-x_{jL})$
$\displaystyle=:J_{1}+J_{2}+J_{3}+J_{4}.$

Denote

$\Omega_{i}=\\{y\in\mathbb{R}^{N}\,\text{such~{}
that}\,\,|y-x_{iL}|\leq|y-x_{jL}|\,\,\text{for~{}all}\,j\neq i\\}.$

Assuming $y\in\Omega_{i}$, then

$\displaystyle|J_{1}|\leq$ $\displaystyle
C\frac{\mu_{iL}^{2}}{(1+\mu_{L}|y-x_{iL}|)^{4}}\sum\limits_{j\neq
i}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}$
$\displaystyle+C\Big{(}\sum\limits_{j\neq
i}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}\Big{)}^{2^{*}-1}.$

Since

$\displaystyle|y-x_{jL}|\geq|y-x_{iL}|,\,\,\forall\,\,\,y\in\Omega_{i},$

we find

$\displaystyle\frac{\mu_{L}^{2}}{(1+\mu_{L}|y-x_{iL}|)^{4}}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}\leq\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{iL}|)^{\frac{N+2}{2}+\tau}}\frac{1}{(\mu_{L}|x_{iL}-x_{jL}|)^{\frac{N+2}{2}-\tau}}.$

On the other hand

$\displaystyle\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}\leq\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\frac{(N-2)\tau}{N+2}}}\frac{1}{(\mu_{L}|x_{jL}-x_{iL}|)^{\frac{N-2}{2}-\frac{\tau(N-2)}{N+2}}},$

we deduce

$\displaystyle\Big{(}\sum\limits_{i\neq
j}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}\Big{)}^{2^{*}-1}\leq\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{iL}|)^{\frac{N+2}{2}+\tau}}\frac{1}{(\mu_{L}|x_{iL}-x_{jL}|)^{\frac{N+2}{2}-\tau}}.$

Thus, we obtain

$\displaystyle\|J_{1}\|_{**}\leq C\sum\limits_{i\neq
j}\frac{1}{(\mu_{L}|x_{jL}-x_{iL}|)^{\frac{N+2}{2}-\tau}}\leq\frac{C}{(\mu_{L}L)^{\frac{N+2}{2}-\tau}}.$

Now, we estimate $J_{2}.$ When $|y-x_{jL}|\geq 1,$ we have

$\displaystyle|J_{2}|\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}$
(A.9) $\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{j}|)^{\frac{N+2}{2}+\tau}}\frac{1}{\mu^{2+\frac{N-6}{2}-\tau}_{L}}$
(A.10) $\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{j}|)^{\frac{N+2}{2}+\tau}}\frac{1}{\mu^{\frac{N-2}{2}-\tau}_{L}}.$
(A.11)

When $|y-x_{jL}|\leq 1,$ we have

$\displaystyle|J_{2}|\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}\xi(y-x_{jL})}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}|y-x_{j}|^{\beta}$
(A.12) $\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}\xi(y-x_{jL})}{(1+\mu_{L}|y-x_{j}|)^{\frac{N+2}{2}+\tau}}\frac{(|x_{jL}-x_{j}|^{\beta}+|y-x_{jL}|^{\beta})}{\mu_{L}^{2}(1+\mu_{L}|y-x_{j}|)^{\frac{N-6}{2}-\tau}}$
(A.13) $\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N+2}{2}+\tau}}\frac{(|x_{jL}-x_{j}|^{\beta}+|y-x_{jL}|^{\beta})}{(1+\mu_{L}|y-x_{j}|)^{\frac{N-2}{2}-\tau}}$
(A.14) $\displaystyle\leq$
$\displaystyle\Big{(}\frac{C}{\mu_{L}^{\frac{N-2}{2}-\tau}}+C\max_{j}{|x_{jL}-x_{j}|^{\beta}}\Big{)}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{j}|)^{\frac{N+2}{2}+\tau}},$
(A.15)

since $\tau<\frac{N-2}{2},\beta-\frac{N-2}{2}+\tau>0$ and

$\frac{|y-x_{j,L}|^{\beta}}{(1+\mu_{L}|y-x_{j,L}|)^{\frac{N-2}{2}-\tau}}\leq\frac{C}{\mu_{L}^{\frac{N-2}{2}-\tau}}.$

Hence it follows from (A.9) and (A.12) that

$\displaystyle\|J_{2}\|_{**}\leq\Big{(}\frac{C}{\mu_{L}^{\frac{N-2}{2}-\tau}}+C\max_{j}{|x_{jL}-x_{j}|^{\beta}}\Big{)}.$

Also, we can estimate directly

$\displaystyle|J_{3}|$ $\displaystyle\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}|\Delta\xi(y-x_{jL})|}{(1+\mu_{L}|y-x_{jL}|)^{N-2}}$
$\displaystyle\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N+2}{2}+\tau}}\frac{|\Delta\xi(y-x_{jL})|}{\mu^{2}_{L}(1+\mu_{L}|y-x_{jL}|)^{\frac{N-6}{2}-\tau}}$
$\displaystyle\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N+2}{2}+\tau}}\frac{1}{\mu^{\frac{N-2}{2}-\tau}_{L}},$

where we use the following fact that

$\frac{1}{(1+\mu_{L}|y-x_{jL}|)}\sim\frac{C}{\mu_{L}},\,\,\,\text{when}\,\,\,1\leq|y-x_{jL}|\leq
2.$

Hence, we have

$\|J_{3}\|_{**}\leq C(\frac{1}{\mu_{L}})^{\frac{N-2}{2}-\tau}.$ (A.16)

Similar to (A.16), we can estimate $\|J_{4}\|_{**}\leq
C(\frac{1}{\mu_{L}})^{\frac{N-2}{2}-\tau}$. Thus, combining all the estimates
above, we complete the proof. ∎

######  Lemma A.7.

If $N\geq 5$, then

$\displaystyle\frac{\partial
I}{\partial\mu_{L}}=m\Big{(}-\frac{B_{1}}{\mu_{L}^{\beta+3}}+\sum\limits_{j=1}^{m}\frac{B_{2}}{\mu_{L}^{N-1}|x_{0L}-x_{jL}|^{N-2}}+o(\frac{1}{\mu_{L}^{\beta+3}})\Big{)}$

where $B_{1},B_{2}$ are constants, and

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+Q(y)u^{2}dx-\frac{1}{2^{*}}\int_{\mathbb{R}^{N}}|u|^{2^{*}}.$

###### Proof.

Firstly, we have

$\displaystyle\frac{\partial
I(W_{\textbf{x},\mu_{L}})}{\partial\mu}=\frac{\partial
I(U_{m})}{\partial\mu_{L}}+O(\frac{m}{\mu_{L}^{N-1}}),$ (A.17)

where $U_{m}=\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}$.

Direct calculations show

$\displaystyle\frac{\partial
I(U_{m})}{\partial\mu_{L}}=\int_{\mathbb{R}^{N}}Q(y)U_{m}\frac{\partial
U_{m}}{\partial\mu_{L}}-\int_{\mathbb{R}^{N}}\Big{(}U_{m}^{2^{*}-1}-\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}^{2^{*}-1}\Big{)}\frac{\partial
U_{m}}{\partial\mu_{L}}+O(\frac{1}{\mu_{L}^{\beta+3}}).$ (A.18)

It is easy to check that

$\displaystyle\int_{\mathbb{R}^{N}}Q(y)U_{m}\frac{\partial
U_{m}}{\partial\mu_{L}}=$ $\displaystyle
m\int_{\mathbb{R}^{N}}Q(y)U_{x_{0L},\mu_{L}}\frac{\partial
U_{x_{0L},\mu_{L}}}{\partial\mu_{L}}+O\Big{(}\frac{1}{\mu_{L}}\int_{\mathbb{R}^{N}}U_{x_{0L},\mu_{L}}\sum\limits_{j=1}^{m}U_{x_{jL},\mu_{L}}\Big{)}$
(A.19) $\displaystyle=$ $\displaystyle
m\int_{\mathbb{R}^{N}}a|y-x_{0}|^{\beta}U_{x_{0},\mu}\frac{\partial
U_{x_{0L},\mu_{L}}}{\partial\mu_{L}}+O\Big{(}\frac{1}{\mu_{L}^{3}}\sum\limits_{i\neq
j}^{m}\frac{1}{(\mu_{L}|x_{iL}-x_{jL}|)^{(N-4)}}\Big{)}$ (A.20)
$\displaystyle=$ $\displaystyle
m\Big{(}\frac{a}{\mu_{L}^{\beta+3}}\int_{\mathbb{R}^{N}}U_{0,1}^{2}+O(\frac{1}{\mu_{L}^{\beta+4}})\Big{)}.$
(A.21)

On the other hand, similar to [26], we obtain

$\displaystyle\int_{\mathbb{R}^{N}}(U_{m}^{2^{*}-1}-\sum\limits_{j=0}^{m}U_{x_{jL},\mu_{L}}^{2^{*}-1})\frac{\partial
U_{m}}{\partial\mu_{L}}=m(-\sum\limits_{j=1}^{m}\frac{B_{2}}{\mu_{L}^{N-1}|x_{0L}-x_{jL}|^{N-2}}+O(\frac{1}{\mu_{L}^{\beta+4}})),$
(A.22)

for some constant $B_{2}>0$, we obtain the result. ∎

######  Lemma A.8.

If $N\geq 5$, then

$\displaystyle\frac{\partial I}{\partial
y_{i}}=m\Big{(}-\frac{C_{1}(x_{jL}-x_{j})_{i}}{\mu_{L}^{\beta+2}}+\sum\limits_{i\neq
j}^{m}\frac{C_{2}}{\mu_{L}^{N-2}|x_{iL}-x_{jL}|^{N-2}}+o(\frac{1}{\mu_{L}^{\beta+2}})\Big{)},\,\,i=1,2,...,N.$

where $C_{1},C_{2}$ are constants.

###### Proof.

Since

$\displaystyle\int_{\mathbb{R}^{N}}Q(y)U_{m}\frac{\partial U_{m}}{\partial
y_{i}}=-\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{\partial Q(y)}{\partial
y_{i}}U_{m}^{2}$ (A.23) $\displaystyle=$
$\displaystyle-\frac{\beta}{2}\int_{\mathbb{R}^{N}}a\beta\big{|}\mu_{L}^{-1}y_{i}+(x_{jL}-x_{j})_{i}\big{|}^{\beta-2}\big{(}\mu_{L}^{-1}y_{i}+(x_{jL}-x_{j})_{i}\big{)}U_{0,1}^{2}$
(A.24)
$\displaystyle+O\Big{(}\frac{1}{\mu_{L}^{\beta+2+\varsigma}}+|x_{jL}-x_{j}|^{\beta+2+\varsigma}\Big{)}$
(A.25) $\displaystyle=$
$\displaystyle\frac{amC}{\mu_{L}^{\beta+2}}(x_{jL}-x_{j})_{i}+O(\frac{1}{\mu_{L}^{\beta+2+\varsigma}}+|x_{jL}-x_{j}|^{\beta+2+\varsigma}),$
(A.26)

combining (A.23) and by the similar argument as that of Lemma A.7, we can
complete the proof. ∎

##  Appendix B Estimate of the error term

We know that $u_{L}$ be a solution of equation (1.1) with the form

$u_{L}=\sum_{j=0}^{m}\xi(y-x_{jL})U_{x_{jL},\mu_{L}}+\varphi_{L},$ (B.1)

and

$-\Delta\varphi_{L}-Q(y)\varphi_{L}-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}=N_{L}(\varphi_{L})+l_{L},$
(B.2)

where $l_{L}$ and $N_{L}(\varphi_{L})$ are defined in (A.4) and (A.6)
respectively. Observe that in the decomposition (B.1), we do not assume that
$\varphi_{L}\in\mathbf{H}_{m}$. See (2.1) for the definition of
$\mathbf{H}_{m}.$ We would like to point out that $x_{jL}$ and
$\tilde{C}_{m}\mu_{L}^{\frac{N-2}{2}}$ may not be a maximum point and the
maximum value of $u_{L}$ in $B_{\delta}(x_{jL}),$ respectively. Let
$\bar{x}_{jL}\in B_{1}(x_{j})$ be such that
$u_{L}(\bar{x}_{jL})=\max_{B_{1}(x_{j})}u_{L}:=\tilde{C}_{m}\bar{\mu}_{L}^{\frac{N-2}{2}}$.
From (1.5), we can deduce that as $L\to+\infty$,
$\bar{\mu}_{L}=\mu_{L}(1+o_{L}(1)),$ and
$\mu_{L}(\bar{x}_{jL}-x_{jL})=o_{L}(1).$ As a result, we have

$\begin{split}&U_{x_{jL},\mu_{L}}-U_{\bar{x}_{jL},\bar{\mu}_{L}}\\\
=&O\bigl{(}|\mu_{L}(\bar{x}_{jL}-x_{jL})|+\mu_{L}^{-1}|\bar{\mu}_{L}-\mu_{L}|\bigr{)}U_{\bar{x}_{jL},\bar{\mu}_{L}}=o_{L}(1)U_{\bar{x}_{jL},\bar{\mu}_{L}},\end{split}$

and

$|\varphi_{L}(x)|=o_{L}(1)\sum_{j=0}^{m}\frac{\bar{\mu}_{L}^{\frac{N-2}{2}}}{(1+\bar{\mu}_{L}|y-\bar{x}_{jL}|)^{\frac{N-2}{2}+\tau}}.$

So, we find that in (B.1), $x_{jL}$ and $\mu_{L}$ can be replaced by
$\bar{x}_{jL}$ and $\bar{\mu}_{L}$ respectively. For simplicity, in the
following, we still use $x_{jL}$ and $\mu_{L}$ to denote $\bar{x}_{jL}$ and
$\bar{\mu}_{L}$ respectively. Thus in (B.1), it holds

$|\varphi_{L}(x_{jL})|=O\Bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}}L^{N-2}}\Bigr{)},\quad|\nabla\varphi_{L}(x_{jL})|=O\Bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}}L^{N-1}}\Bigr{)}.$
(B.3)

In the following, we will use (B.2) to estimate $\omega_{L}$.

######  Proposition B.1.

Assume that $N\geq 5.$ There holds

$\|\varphi_{L}\|_{*}\leq
C\Big{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}-\tau}}+\max\limits_{i}|x_{iL}-x_{i}|^{\beta}\Big{)}.$

###### Proof.

Since

$\displaystyle-\Delta\varphi_{L}+Q(y)\varphi_{L}-(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}=N_{L}(\varphi_{L})+l_{L},$

noting that $Q(y)\geq 0,$ we have

$\displaystyle|\varphi_{L}|\leq$ $\displaystyle
C\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}|\varphi_{L}|dz+\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}(|N(\varphi_{L})|+|l_{L}|).$
(B.4)

We can check easily

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}(N_{L}(\varphi_{L})+l_{L})\Big{|}$
(B.5) $\displaystyle\leq
C(\|\varphi_{L}\|_{*}^{\min\\{2^{*}-1,2\\}}+\|l_{L}\|_{**})\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N+2}{2}}}{(1+\mu_{L}|z-x_{jL}|)^{\frac{N+2}{2}+\tau}}$
(B.6) $\displaystyle\leq$ $\displaystyle
C(\|\varphi_{L}\|_{*}^{\min\\{2^{*}-1,2\\}}+\|l_{L}\|_{**})\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{jL}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}.$
(B.7)

Suppose $N\geq 6$, we have

$\displaystyle W_{\textbf{x},\mu_{L}}^{2^{*}-2}\leq
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{2}}{(1+\mu_{L}|z-x_{jL}|)^{4}},$

If $N=5,$ then by discrete Hölder inequality we have

$\displaystyle W_{\textbf{x},\mu_{L}}^{2^{*}-2}$ $\displaystyle\leq
C\mu^{2}_{L}\Big{(}\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{3}}\Big{)}^{\frac{4}{3}}$
$\displaystyle\leq
C\mu^{2}_{L}\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{4-\frac{\tau}{3}}}\Big{(}\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{\tau}}\Big{)}^{\frac{1}{3}}$
$\displaystyle\leq
C\mu^{2}_{L}\sum\limits_{j=0}^{m}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{4-\frac{\tau}{3}}}.$

So, we obtain that for $N\geq 6$,

$\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{|y-z|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|z-x_{jL}|)^{\frac{N-2}{2}+\tau}}dz$
$\displaystyle\leq$
$\displaystyle\mu_{L}^{\frac{N+2}{2}}\sum\limits_{j=0}^{m}\int_{\mathbb{R}^{N}}\frac{1}{|y-z|^{N-2}}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{\frac{N-2}{2}+\tau+4}}dz$
$\displaystyle+\mu_{L}^{\frac{N+2}{2}}\sum\limits_{j=0}^{m}\sum\limits_{i\neq
j}\int_{\mathbb{R}^{N}}\frac{1}{|y-z|^{N-2}}\frac{1}{(1+\mu_{L}|z-x_{jL}|)^{4}}\frac{1}{(1+\mu_{L}|z-x_{iL}|)^{\frac{N-2}{2}+\tau}}dz$
$\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{2+\frac{N-2}{2}}},$

which implies that if $N\geq 6,$

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}dz\Big{|}\leq\|\varphi_{L}\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{2+\frac{N-2}{2}}}$
(B.8)

Also, by the similar argument we can show that for $N=5$, there is a
$\varsigma>0$ such that

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}\frac{1}{|z-y|^{N-2}}W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}dz\Big{|}\leq
C\|\varphi_{L}\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau+\varsigma}}.$
(B.9)

Therefore, it follows from (B.4) to (B.9) that

$\displaystyle|\varphi_{L}|\leq$ $\displaystyle
C(\|\varphi_{L}\|_{*}^{\min\\{2^{*}-1,2\\}}+\|l_{L}\|_{**})\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}$
(B.10)
$\displaystyle+C\|\varphi_{L}\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{L}^{\frac{N-2}{2}}}{(1+\mu_{L}|y-x_{jL}|)^{\frac{N-2}{2}+\tau+\varsigma}}$
(B.11)

where $\varsigma>0$ is a constant.

Assume that there exists
$y\in\mathbb{R}^{N}\setminus\cup_{j=0}^{m}B_{R\mu_{L}^{-1}}(x_{jL})$ for some
large $R>0$, such that $\|\omega\|_{*}$ is achieved at $y$. Then, (B.10)
yields

$\|\varphi_{L}\|_{*}\leq
C\bigl{(}\|\varphi_{L}\|_{*}^{\min(2^{*}-1,2)}+\|l_{L}\|_{**}\bigr{)}+o_{R}(1)\|\varphi_{L}\|_{*}.$
(B.12)

Noting that $\|\varphi_{L}\|_{*}\to 0$ as $L\to+\infty$, we have

$\|\varphi_{L}\|_{*}\leq C\|l_{L}\|_{**}.$ (B.13)

Suppose that $\|\varphi_{L}\|_{*}$ is achieved at $y\in
B_{R\mu_{L}^{-1}}(x_{jL}).$ Let

$\tilde{\varphi}_{L}(y)=\mu_{L}^{-\frac{N-2}{2}}\varphi_{L}(\mu_{L}^{-1}y+x_{jL}),$

and

$|||\tilde{\varphi}_{L}|||_{*}=\sup_{y\in\mathbb{R}^{N}}\Bigl{(}\sum_{i=0}^{m}\frac{\mu_{L}^{-\frac{N-2}{2}}\mu_{L}^{\frac{N-2}{2}}}{(1+\frac{\mu_{L}}{\mu_{L}}|y-\mu_{L}(x_{iL}-x_{jL})|)^{\frac{N-2}{2}+\tau}}\Bigr{)}^{-1}|\tilde{\varphi}_{L}(y)|.$

Then $|||\tilde{\omega}_{L}|||_{*}$ is achieved at some $y\in B_{R}(0)$.

Suppose that

$||\varphi_{L}||_{*}\geq
n_{L}\Bigl{(}\frac{1}{\mu_{L}^{\frac{N-2}{2}-\tau}}+C\max_{i}|x_{iL}-x_{i}|^{\beta}\Bigr{)},$

for some $n_{L}\to\infty$. Then as $L\to+\infty$,
$\eta_{L}=\frac{\tilde{\varphi}_{L}}{|||\tilde{\varphi}_{L}|||_{*}}$ converges
to $\eta\neq 0$, which satisfies

$-\Delta\eta-(2^{*}-1)U_{0,1}^{2^{*}-2}\eta=0,\quad\text{in}\;\mathbb{R}^{N},$

since
$\frac{\|l_{L}\|_{**}}{||\tilde{\varphi}_{L}||_{*}}\leq\frac{C}{n_{L}}\to 0$.
This gives

$\eta=\alpha_{0}\frac{\partial
U_{0,\mu_{L}}}{\partial\mu_{L}}\bigr{|}_{\mu_{L}=1}+\sum_{j=1}^{N}\alpha_{j}\frac{\partial
U_{0,1}}{\partial y_{j}},$

for some constant $\alpha_{j}.$

On the other hand, we have

$\displaystyle\tilde{\varphi}_{L}(0)$ $\displaystyle=$
$\displaystyle\mu_{L}^{-\frac{N-2}{2}}\varphi_{L}(x_{jL})=O\bigl{(}\frac{1}{\mu_{L}^{N-2}L^{N-2}}\bigr{)},$
$\displaystyle\nabla\tilde{\varphi}_{L}(0)$ $\displaystyle=$
$\displaystyle\mu_{L}^{-\frac{N}{2}}\nabla\varphi_{L}(x_{jL})=O\bigl{(}\frac{1}{\mu_{L}^{N-1}L^{N-1}}\bigr{)}.$

So, we find $\eta(0)=0$ and $\nabla\eta(0)=0$, which implies
$\alpha_{0}=\alpha_{1}=\cdots=\alpha_{N}=0$. This is a contradiction.

∎

######  Corollary B.2.

For any $\delta>0$, it holds

$\displaystyle|\varphi_{L}|,|\nabla\varphi_{L}|\leq\frac{C}{\mu_{L}^{\tau}}(\frac{1}{\mu_{L}^{\frac{N-2}{2}-\tau}}+\max\limits_{i}|x_{iL}-x_{i}|^{\beta}).$

###### Proof.

It follows from Proposition B.1 that

$\displaystyle|\varphi_{L}|\leq$
$\displaystyle\|\varphi_{L}\|_{*}\sum\limits_{j=0}^{m}\frac{\mu_{jL}^{\frac{N-2}{2}}}{(1+\mu_{jL}|y-x_{jL}|)^{\frac{N-2}{2}+\tau}}$
$\displaystyle\leq$
$\displaystyle\frac{C}{\mu^{\tau}}(\frac{1}{\mu_{L}^{\frac{N-2}{2}-\tau}}+\max\limits_{i}|x_{iL}-x_{i}|^{\beta}),\,\,\,\,x\in
B_{4\delta}(x_{jL})\setminus B_{\frac{\delta}{2}}(x_{jL}).$

On the other hand, using the $L^{p}$ estimates, we can deduce that for any
$p>1$,

$\displaystyle\|\varphi_{L}\|_{W^{2p}(B_{4\delta}(x_{jL})\setminus
B_{\frac{\delta}{2}}(x_{jL}))}\leq$ $\displaystyle
C\|\varphi_{L}\|_{L^{\infty}(B_{4\delta}(x_{jL})\setminus
B_{\frac{\delta}{2}}(x_{jL}))}$
$\displaystyle+C\|(2^{*}-1)W_{\textbf{x},\mu_{L}}^{2^{*}-2}\varphi_{L}+N_{L}(\varphi_{L})+l_{L}\|_{L^{\infty}(B_{4\delta}(x_{jL})\setminus
B_{\frac{\delta}{2}}(x_{jL}))}$ $\displaystyle\leq$ $\displaystyle
C\|\varphi_{L}\|_{L^{\infty}(B_{4\delta}(x_{jL})\setminus
B_{\frac{\delta}{2}}(x_{jL}))}+\frac{C}{\mu_{L}}(\|N_{L}(\varphi_{L})\|_{**}+\|l_{L}\|_{**})$
$\displaystyle\leq$
$\displaystyle\frac{C}{\mu_{L}^{\tau}}(\frac{1}{\mu_{L}^{\frac{N-2}{2}-\tau}}+\max\limits_{i}|x_{iL}-x_{i}|^{\beta}),$

the results follow from Lemma A.6 and Proposition B.1. ∎

##  Appendix C Proof of (3.9)

In this section, we mainly prove (3.9) which involve some technical and
precise computations.

###### Proof of (3.9).

For notations of simplicity, we may denote
$\textbf{d}=\max_{i}|x_{iL}-x_{i}|.$ Recall that
$\delta=\mu^{-\theta}_{L},\theta$ is defined in (3.6). By the assumption
$(Q_{3}),$ we have

$\displaystyle\int_{\partial B_{\delta}(x_{jL})}Q(y)u_{L}^{2}\nu_{i}$
$\displaystyle=$ $\displaystyle\int_{\partial
B_{\delta}(x_{jL})}\big{(}a|y-x_{j}|^{\beta}+O(|y-x_{j}|^{\beta+1})\big{)}u_{L}^{2}\nu_{i}$
$\displaystyle=$ $\displaystyle\int_{\partial
B_{\delta}(x_{jL})}\big{(}a(|y-x_{jL}|^{\beta}+|x_{jL}-x_{j}|^{\beta})+O(|y-x_{jL}|^{\beta-1}|x_{jL}-x_{j}|+|y-x_{jL}||x_{jL}-x_{j}|^{\beta-1}$
$\displaystyle\quad\quad+|y-x_{j}|^{\beta+1})\big{)}u_{L}^{2}\nu_{i}$
$\displaystyle=$ $\displaystyle a\int_{\partial
B_{\delta}(x_{jL})}|y-x_{jL}|^{\beta}\Big{(}\sum_{i\neq
j}^{m}\xi(y-x_{iL})U_{\mu_{L},x_{iL}}+\varphi_{L}\Big{)}^{2}\nu_{i}$
$\displaystyle\quad+O\Big{(}\int_{\partial
B_{\delta}(x_{jL})}\big{(}\textbf{d}^{\beta}+\delta^{\beta-1}\textbf{d}+\delta\textbf{d}^{\beta-1}+\delta^{\beta+1}\big{)}u_{L}^{2}\Big{)}$
$\displaystyle=:$ $\displaystyle F_{1}+F_{2}$ $\displaystyle=$ $\displaystyle
O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$

since by Lemma A.4, Proposition B.1 and (3.6), we have

$\displaystyle F_{1}$ $\displaystyle\leq$ $\displaystyle C\int_{\partial
B_{\delta}(x_{jL})}\delta^{\beta}\Big{\\{}\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu^{2}_{L}|y-x_{iL}|^{2})^{\frac{N-2}{2}}}\big{)}^{2}+\|\varphi\|^{2}_{*}\big{(}\sum_{i=0}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|y-x_{iL}|)^{\frac{N-2}{2}+\tau}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\int_{\partial
B_{\delta}(x_{jL})}\delta^{\beta}\Big{\\{}\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{\mu^{2}_{L}|x_{jL}-x_{iL}|^{2})^{\frac{N-2}{2}}}\big{)}^{2}+\|\varphi\|^{2}_{*}$
$\displaystyle\quad\quad\big{(}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{N-2+2\tau}}+\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|x_{jL}-x_{iL}|)^{\frac{N-2}{2}+\tau}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\int_{\partial
B_{\delta}(x_{jL})}\delta^{\beta}\Big{\\{}\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}+\|\varphi_{L}\|^{2}_{*}\big{(}\frac{1}{\mu^{2\tau}_{L}\delta^{N-2+2\tau}}+\frac{1}{\mu^{2\tau}_{L}L^{N-2+2\tau}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\delta^{\beta+N-1}\Big{\\{}\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}+\big{(}\frac{1}{\mu^{\frac{N-2}{2}-\tau}_{L}}+\textbf{d}^{\beta}\big{)}^{2}\big{(}\frac{1}{\mu^{2\tau}_{L}\delta^{N-2+2\tau}}+\frac{1}{\mu^{2\tau}_{L}L^{N-2+2\tau}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\delta^{\beta+N-1}\Big{(}\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}+\frac{1}{\mu^{N-2}_{L}\delta^{N-2+2\tau}}+\frac{\textbf{d}^{2\beta}}{\mu^{2\tau}_{L}\delta^{N-2+2\tau}}\Big{)}$
$\displaystyle=$ $\displaystyle
O\Big{(}\frac{1}{\mu^{N-2+2(\beta+4-N)+(\beta+N-1)\theta}_{L}}+\frac{1}{\mu^{N-2+(\beta+1-2\tau)\theta}_{L}}+\frac{\textbf{d}^{2\beta}}{\mu^{2\tau+(\beta+N-1)\theta}_{L}}\Big{)}$
$\displaystyle=$ $\displaystyle O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)}$

and

$\displaystyle F_{2}$ $\displaystyle\leq$ $\displaystyle C\int_{\partial
B_{\delta}(x_{jL})}\big{(}\textbf{d}^{\beta}+\delta^{\beta-1}\textbf{d}+\delta\textbf{d}^{\beta-1}+\delta^{\beta+1}\big{)}\Big{(}\sum_{i=0}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|y-x_{iL}|)^{N-2}}\Big{)}^{2}$
(C.1) $\displaystyle=:$ $\displaystyle
F_{2,1}+F_{2,2}+F_{2,3}+F_{2,4}=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$
(C.2)

where in (C.1) we use the following estimates

$\displaystyle F_{2,1}$ $\displaystyle\leq$ $\displaystyle
C\textbf{d}^{\beta}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{2(N-2)}}+\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|x_{jL}-x_{iL}|)^{N-2}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\textbf{d}^{\beta}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{N-2}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\textbf{d}^{\beta}\Big{(}\frac{1}{\delta^{N-2}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{)}\delta^{N-1}$
$\displaystyle=$ $\displaystyle
C\textbf{d}^{\beta}\Big{(}\frac{1}{\mu^{\theta}_{L}}+\frac{1}{\mu^{N-2+(N-1)\theta}_{L}L^{2(N-2)}}\Big{)}=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$

$\displaystyle F_{2,2}$ $\displaystyle\leq$ $\displaystyle
C\delta^{\beta-1}\textbf{d}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{2(N-2)}}+\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|x_{jL}-x_{iL}|)^{N-2}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\delta^{\beta-1}\textbf{d}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{N-2}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\delta^{\beta-1}\textbf{d}\Big{(}\frac{1}{\delta^{N-2}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{)}\delta^{N-1}$
$\displaystyle=$ $\displaystyle
C\textbf{d}\Big{(}\frac{1}{\mu^{\beta\theta}_{L}}+\frac{1}{\mu^{N-2+(N+\beta-2)\theta}_{L}L^{2(N-2)}}\Big{)}=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$

$\displaystyle F_{2,3}$ $\displaystyle\leq$ $\displaystyle
C\delta\textbf{d}^{\beta-1}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{2(N-2)}}+\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|x_{jL}-x_{iL}|)^{N-2}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\delta\textbf{d}^{\beta-1}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{N-3}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\delta\textbf{d}^{\beta-1}\Big{(}\frac{\mu_{L}}{\delta^{N-3}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{)}\delta^{N-1}$
$\displaystyle=$ $\displaystyle
C\textbf{d}^{\beta-1}\Big{(}\frac{1}{\mu^{3\theta-1}_{L}}+\frac{1}{\mu^{N-2+(N+\beta-2)\theta}_{L}L^{2(N-2)}}\Big{)}$
$\displaystyle=$ $\displaystyle O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)},$

and

$\displaystyle F_{2,4}$ $\displaystyle\leq$ $\displaystyle
C\delta^{\beta+1}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{\mu^{N-2}_{L}}{(1+\mu_{L}\delta)^{2(N-2)}}+\big{(}\sum_{i\neq
j}^{m}\frac{\mu^{\frac{N-2}{2}}_{L}}{(1+\mu_{L}|x_{jL}-x_{iL}|)^{N-2}}\big{)}^{2}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C\delta^{\beta+1}\int_{\partial
B_{\delta}(x_{jL})}\Big{\\{}\frac{1}{\mu_{L}^{N-2}\delta^{2(N-2)}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{\\}}$
$\displaystyle\leq$
$\displaystyle\delta^{\beta+1}\Big{(}\frac{1}{\mu_{L}^{N-2}\delta^{2(N-2)}}+\frac{1}{\mu^{N-2}_{L}L^{2(N-2)}}\Big{)}\delta^{N-1}$
$\displaystyle=$ $\displaystyle
C\Big{(}\frac{1}{\mu^{N-2+(\beta+4-N)\theta}_{L}}+\frac{1}{\mu^{N-2+(\beta+4-N)\theta}_{L}L^{2(N-2)}}\Big{)}=O\Big{(}\frac{1}{\mu^{\beta+2}_{L}}\Big{)}.$

∎

Acknowledgements The authors would like to thank Professor Shusen Yan for the
helpful discussion with him. This paper was supported by NSFC grants
(No.12071169) and the Fundamental Research Funds for the Central
Universities(No.KJ02072020-0319).

## References

  * [1] A. Bahri, J. Coron, The scalar curvature problem on the standard three dimentional sphere, J. Funct. Anal., 95(1991), 106-172.
  * [2] T. Bartsch, T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math., 571(2004), 131-143.
  * [3] T. Bartsch, T. Weth, M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology of the domain, Calc. Var. Partial Differential Equations, 18(2003), 253-268.
  * [4] V. Benci, G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $\mathbb{R}^{N}.$ J. Funct. Anal., 88(1990), 90-117.
  * [5] H. Brezis, Y. Li, Some nonlinear elliptic equations have only constant solutions. J. Partial Differ. Equ., 19(2006), 208-217.
  * [6] H. Brezis, L. Peletier, Elliptic equations with critical exponent on spherical caps of $S^{3}$. J. Anal. Math., 98(2006), 279-316.
  * [7] D. Cao, H.Heniz, Uniqueness of positive multi-bump bound states of nonlinear Schrödinger equations, Math. Z., 243(2003), 599-642.
  * [8] D. Cao, E. Noussair, S.Yan, Existence and uniquenesss results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15(1998),73-111.
  * [9] W. Chen, J. Wei, S. Yan, Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 252(2012), 2425-2447.
  * [10] S.Chang, P. Yang, A perturbation result in prescribing scalar curvature on $S^{n}$, Duke Math. J., 64(1991), 27-69.
  * [11] Y. Deng, C.-S. Lin, S. Yan, On the prescribed scalar curvature problem in $\mathbb{R}^{N}$, local uniqueness and periodicity, J. Math Pure Appl.,104(2015), 1013-1044.
  * [12] 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.
  * [13] O. Druet, From one bubble to several bubbles: the low-dimensional case. J. Differential Geom., 63(2003), 399-473.
  * [14] W. Ding, W. Ni, On the elliptic equation $\Delta u+Ku^{\frac{N+2}{N-2}}=0$ and related topics, Duck math. J., 52(1985), 485-506.
  * [15] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 34(1981), 525-598.
  * [16] L.Glangetas, Uniqueness of positive solutions of a nonlinear elliptic equation involving the critical exponent, Nonlin. Anal. TMA, 20(1993), 571-603.
  * [17] M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19(2002), 261-280.
  * [18] Y. Guo, S. Peng, S. Yan, Local uniqueness and periodicity induced by concentration. Proc. Lond. Math. Soc., (3) 114(2017), 1005-1043.
  * [19] Q. He, C. Wang, Q. Wang, A new type of bubble solutions for a Schrödinger equation with critical growth, arXiv:2101.03284.
  * [20] Y. Y. Li, On $-\Delta u=K(x)u^{5}$ in $\mathbb{R}^{3}$, Comm. Pure Appl. Math., 46(1993), 303-340.
  * [21] Y. Y. Li, Prescribing scalar curvature on $S^{3};S^{4},$ and related problems, J. Funct. Anal., 118(1993), 43-118.
  * [22] Y. Y. Li, Prescribing scalar curvature on $S^{n}$ and related problems. PartI, J. Differential Equations, 120(1995), 319-410.
  * [23] Y. Y. Li, J. Wei, H. Xu, Multi-bump solutions of $-\Delta u=K(x)u^{\frac{n+2}{n-2}}$ on lattices in $\mathbb{R}^{n},$ J. Reine Angew. Math., 743(2018), 163-211.
  * [24] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1(1984), 109-145.
  * [25] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
  * [26] S. Peng, C. Wang, S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal., 274(2018), 2606-2633.
  * [27] J. Wei and Y. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^{N}$, J. Funct. Anal., 258(2010), 3048-3081.
  * [28] H. Xu, Critical exponent elliptic equations: Gluing via the implicit function theorem and the moving sphere method, Ph. D. thesis, 2007.
  * [29] S. Yan, Concentration of solutions for the scalar curvature equation on $\mathbb{R}^{N}$, J. Differential Equations, 163(2000), 239-264.