Nondegeneracy of the solutions for elliptic problem with critical exponent

Abstract

This paper deals with the following nonlinear elliptic equation:

$$ -\Delta u=Q(|y'|,y'')u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,{ \mathbb{R}}^{N},\,\,u\in D^{1,2}({\mathbb{R}}^{N}), $$

where $(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}$, $N\geq 5$, $Q(|y'|,y'')$ is a bounded nonnegative function in $\mathbb{R}^{2}\times {\mathbb{R}}^{N-2}$. By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).

1 Introduction and the main results

In this paper, we consider the following problem:

$$\begin{aligned} -\Delta u=Q(|y'|,y'')u^{\frac{N+2}{N-2}}, \, u>0,\, u\in D^{1,2}({ \mathbb{R}}^{N}), \end{aligned}$$

(1.1)

where $Q(y)\geq 0$ and $N\geq 5$. We prove a nondegeneracy result for the positive multibubbling solutions constructed in [14], which is of great independent interest. In the last three decades, there has been considerable interest in the existence and multiplicity of solutions for problem (1.1) under some suitable assumptions on the function $Q(y)$; see, for example, [1, 6, 10, 19] and references therein. When $Q(y)$ is positive and periodic, by gluing approximation solutions into genuine solutions, which concentrate at some isolated maximum points of the function $Q(y)$, Li [10–12] proved that (1.1) has infinitely many multibubbling solutions for $N\geq 3$. When $Q(y)$ is a positive radial function with a strict local maximum at $|x|=r_{0}>0$, Wei and Yan [18] constructed solutions for (1.1) with large number of bubbles concentrating near the sphere $|x|=r_{0}$. Very recently, Li, Wei, and Xu [13] proved the existence of infinitely many bubble solutions for problem (1.1). For the other related problems with critical exponents, we refer to [2–5, 7, 8, 13, 16, 17, 20, 21] and references therein.

It is well known that the functions

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

are the only solutions to the problem

$$ -\Delta u=u^{\frac{N+2}{N-2}},\,\,\,u>0\,\,\,\text{in}\,\,{\mathbb{R}}^{N}. $$

(1.2)

For any point $y\in {\mathbb{R}}^{N}$, we set $y=(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}$ and define

$$ \begin{aligned} H_{s}=\Bigl\{ u:u\in & D^{1,2}({\mathbb{R}}^{N}),\;\; u(y_{1}, -y_{2},y'')= u(y_{1}, y_{2}, y''), \\ & u(r\cos \theta ,r\sin \theta ,y'')=u\Bigl(r\cos (\theta + \frac{2\pi j }{k}),r\sin (\theta +\frac{2\pi j }{k}),y'')\Bigl\} . \end{aligned} $$

Let

$$ x_{j}=\Bigl(\bar{r} \cos \frac{2(j-1)\pi}{k},\bar{r} \sin \frac{2(j-1)\pi}{k},\bar{y}''\Bigr),\,\,\,j=1,2,\ldots ,k, $$

where $\bar{y}''$ is a vector in ${\mathbb{R}}^{N-2}$, and k is an integer.

Let $\delta >0$ be a small constant such that $r^{2}Q(r, y'')>0$ if $|(r,y'')- (r_{0}, y_{0}'')|\le 10\delta$. Let $\zeta (y)= \zeta (|y'|, y'')$ be a smooth function satisfying $\zeta =1$ if $|(r,y'')- (r_{0}, y_{0}'')|\le \delta$, $\zeta =0$ if $|(r,y'')- (r_{0}, y_{0}'')|\ge 2\delta$, and $0\le \zeta \le 1$. Define

$$\begin{aligned} Z_{x_{j},\mu}(y)=\zeta U_{x_{j},\mu},\,\,\,Z^{*}_{\bar{r},\bar{y}'', \mu}=\sum _{j=1}^{k} U_{x_{j},\mu},\;\;\; Z_{\bar{r},\bar{y}'',\mu}(y)= \sum _{j=1}^{k}Z_{x_{j},\mu}(y). \end{aligned}$$

We also assume that $Q(y)$ satisfies the following condition:

$(Q_{1})$ The function $r^{2}Q(r,y'')$ has a critical point $(r_{0},y''_{0})$ satisfying $r_{0}>0$ and $Q(r_{0},y''_{0})=1$, and $\deg (\nabla (r^{2}Q(r,y'')),(r_{0},y''_{0}))\neq 0$.

$(Q_{2})$ $Q(r,y'')\in C^{3}(B_{\vartheta}(r_{0},y_{0}''))$ and $\Delta Q(r_{0},y_{0}''):= \frac{\partial ^{2}Q(r_{0},y_{0}'')}{\partial r^{2}}+\sum \limits _{i=3}^{N} \frac{\partial ^{2}Q(r_{0},y_{0}'')}{\partial y_{i}^{2}}<0$, where $\vartheta >0$ is small.

Let

$$ \|u\|_{*}=\sup _{y\in {\mathbb{R}}^{N}}\Big(\sum _{j=1}^{k} \frac{1}{(1+\mu _{k}|y-x_{k,j}|)^{\frac{N-2}{2}+\tau}}\Big)^{-1}\mu _{k}^{- \frac{N-2}{2}}|u(y)| $$

(1.3)

and

$$ \|f\|_{**}=\sup _{y\in {\mathbb{R}}^{N}}\Big(\sum _{j=1}^{k} \frac{1}{(1+\mu _{k}|y-x_{k,j}|)^{\frac{N+2}{2}+\tau}}\Big)^{-1}\mu _{k}^{- \frac{N+2}{2}}|f(y)|, $$

(1.4)

where $x_{k,j}=(r_{k}\cos \frac{2(j-1)\pi}{k},r_{k}\sin \frac{2(j-1)\pi}{k},x'')$, $\tau \in (\frac{N-4}{N-2},1+\theta )$ is a fixed number, and $\theta >0$ is a small constant.

The result obtained in [14] states the following:

Theorem 1.1

Let $Q\ge 0$ be a bounded function belonging to $C^{1}$. If $Q(|y'|,y'')$ satisfies $(Q_{1})$ and $(Q_{2})$ and $N\ge 5$, then there exists a positive integer $k_{0}>0$ such that for any integer $k\geq k_{0}$, (1.1) has a solution $u_{k}$ of the form

$$\begin{aligned} u_{k}=Z_{\bar{r}_{k},\bar{y}_{k}'',\mu _{k},}+\varphi _{k}=\sum _{j=1}^{k} \zeta U_{x_{j},\mu _{k}}+\varphi _{k}, \end{aligned}$$

(1.5)

where $\varphi _{k}\in H_{s}$. Moreover, as $k\rightarrow +\infty$, $\mu _{k}\in [L_{0}k^{\frac{N-2}{N-4}},L_{1}k^{\frac{N-2}{N-4}}]$, $(\bar{r}_{k},\bar{y}_{k}'')\to (r_{0},y''_{0})$, and $\mu _{k}^{-\frac{N-2}{2}}\|\varphi _{k}\|_{L^{\infty}}\rightarrow 0$, $L_{0}\leq L_{1}$.

Deng, Lin, and Yan [8] studied the periodicity of the bubbling solutions obtained by Li, Wei, and Xu [13]. In this paper, we introduce a new method, which was first introduced by Guo et al. [9], who studied the nondegeneracy of the equation with radial potential function. In our paper, we suppose that $Q(y)$ is partial radial, which makes it different in dealing with the local Pohozaev identities. When $Q(y)$ is partial radial, there appear more variables, which makes the calculation process more complex.

Recall the linearized operator around $u_{k}$ defined by

$$\begin{aligned} L_{k}\eta =-\Delta \eta -(2^{*}-1)Q(y)u_{k}^{2^{*}-2}\eta . \end{aligned}$$

(1.6)

We will show the nondegeneracy of the multibubbling solutions in Theorem 1.1. The main results of our paper is the following.

Theorem 1.2

Suppose that $Q(y)$ satisfies $(Q_{1})$, $(Q_{2})$, and $(\tilde{Q})$: $det(A_{i,l})_{(N-1)\times (N-1)}\neq 0,\,\,i,l=1,2,\ldots,N-1$, $N\geq 5$, where

$$ A_{i,l}= \left \{ \textstyle\begin{array}{l@{\quad}l} \Big( \frac{\partial ^{2}Q}{\partial r^{2}}-\big( \frac{\frac{\partial \Delta Q}{\partial y_{1}}}{2\Delta Q} -1\big)(r \frac{\partial ^{2}Q}{\partial r^{2}}+\sum \limits _{j=3}^{N}y_{j} \frac{\partial ^{2}Q}{\partial r\partial y_{j}}\Big)(r_{0},y''_{0}) \\ \quad \textit{when}\,\,i=l=1, \vspace{0.12cm} \\ \Big( \frac{\partial ^{2}Q}{\partial r\partial y_{l+1}}-\big( \frac{\frac{\partial \Delta Q}{\partial y_{1}}}{2\Delta Q} -1\big)(r \frac{\partial ^{2}Q}{\partial r^{2}}+\sum \limits _{j=3}^{N}y_{j} \frac{\partial ^{2}Q}{\partial r\partial y_{j}}\Big)(r_{0},y''_{0}) \\ \quad \textit{when}\,\,i=1,l=2,3,\ldots,N-1, \vspace{0.12cm} \\ \cos \frac{2\pi i}{k}\Big( \frac{\partial ^{2}Q}{\partial r\partial y_{i+1}}-\big( \frac{\frac{\partial \Delta Q}{\partial y_{1}}}{2\Delta Q} -1\big)(r \frac{\partial ^{2}Q}{\partial r^{2}}+\sum \limits _{j=3}^{N}y_{j} \frac{\partial ^{2}Q}{\partial r\partial y_{j}}\Big)(r_{0},y''_{0}) \\ \quad \textit{when}\,\,l=1,i=2,3,\ldots,N-1, \vspace{0.12cm} \\ \Big[\frac{\partial ^{2}Q}{\partial y_{i+1}\partial y_{l+1}}- \big( \frac{\frac{\partial \Delta Q}{\partial y_{i+1}}}{2\Delta Q} -1\big) (r \frac{\partial ^{2}Q}{\partial r^{2}}+\sum \limits _{j=3}^{N} \frac{\partial ^{2}Q}{\partial y_{j}\partial y_{l+1}}) \Big](r_{0},y''_{0}) \\ \quad \textit{when}\,\,i,l=2,3,\ldots,N-1. \end{array}\displaystyle \right . $$

Let $\eta \in H_{s}$ be a solution of $L_{k}\eta =0$. Then $\eta =0$.

Remark 1.3

If $Q(y)$ is radial, then assumption $(\tilde{Q})$ is just

$$ (\tilde{Q})\,\,\,\,\,\,\,\,\,\, \Delta Q-\Big(\Delta Q+\frac{1}{2}( \Delta Q)'\Big)r\neq 0 \,\,\,\text{at}\,\,\, r=r_{0}. $$

In Sect. 2, we prove the nondegeneracy result by some local Pohozaev identities, which is very crucial in constructing solutions by applying the reduction method.

2 Nondegeneracy of solutions

Let

$$\begin{aligned} -\Delta u=Q(|y'|,y'')u^{2^{*}-1} \end{aligned}$$

(2.1)

and

$$\begin{aligned} -\Delta \eta =(2^{*}-1)Q(|y'|,y'')u^{2^{*}-2}\eta . \end{aligned}$$

(2.2)

Let Ω be a smooth domain in ${\mathbb{R}}^{N}$. Using the Pohozaev identities, we have the identities

$$\begin{aligned} &-\int _{\partial \Omega}\frac{\partial u}{\partial \nu} \frac{\partial \eta}{\partial y_{i}}-\int _{\partial \Omega} \frac{\partial \eta}{\partial \nu}\frac{\partial u}{\partial y_{i}} + \int _{\partial \Omega}\langle \nabla u,\nabla \eta \rangle \nu _{i}+ \int _{\partial \Omega}Qu^{2^{*}-1}\eta \nu _{i} -\int _{\partial \Omega}u^{2^{*}-1}\eta \nu _{i} \\ =&-\int _{\Omega}\frac{\partial Q}{\partial y_{i}}u^{2^{*}-1}\eta \end{aligned}$$

(2.3)

and

$$\begin{aligned} &\int _{\Omega}u^{2^{*}-1}\eta \langle \nabla Q,y-x_{0}\rangle =\int _{ \partial \Omega}Q(y)u^{2^{*}-1}\eta \langle \nu ,y-x_{0}\rangle + \int _{\partial \Omega}\frac{\partial u}{\partial \nu}\langle \nabla \eta ,y-x_{0}\rangle \\ &\,\,\,\,\,\,+\int _{\partial \Omega} \frac{\partial \eta}{\partial \nu}\langle \nabla u,y-x_{0}\rangle - \int _{\partial \Omega}\langle \nabla u,\nabla \eta \rangle \langle \nu ,y-x_{0}\rangle \\ &\,\,\,\,\,\,+\frac{N-2}{2}\int _{\partial \Omega}\eta \frac{\partial u}{\partial \nu}+\frac{N-2}{2}\int _{\partial \Omega}u \frac{\partial \eta}{\partial \nu}. \end{aligned}$$

(2.4)

We define the linear operator

$$\begin{aligned} L_{k}\eta =-\Delta \eta -(2^{*}-1)Q(y)u_{k}^{2^{*}-2}\eta . \end{aligned}$$

Lemma 2.1

There exists a positive constant C such that

$$\begin{aligned} |u_{k}(y)|\leq C\sum \limits _{j=1}^{k} \frac{\mu _{k}^{\frac{N-2}{2}}}{1+(\mu _{k}|y-x_{k,j}|)^{N-2}}\,\, \textit{for all}\,\,\,y\in {\mathbb{R}}^{N}. \end{aligned}$$

Proof

Since the proof is the same as that of Lemma 2.2 of [9], here we omit it. □

We now prove Theorem 1.2 by an indirect method. Suppose that there are $k_{m}\rightarrow +\infty$ satisfying $\|\eta _{m}\|_{*}=1$ and

$$ L_{k_{m}}\eta _{m}=0. $$

(2.5)

Let

$$ \widetilde{\eta}_{m}(y)=\mu _{k_{m}}^{-\frac{N-2}{2}}\eta _{m}(\mu _{k_{m}}^{-1}y+x_{k_{m},1}). $$

(2.6)

Lemma 2.2

We have $\widetilde{\eta}_{m}\rightarrow b_{0}\psi _{0}+\sum \limits _{i=1,i \neq 2}^{N}b_{i}\psi _{i}$ uniformly in $C^{1}(B_{R}(0))$ for any $R>0$, where $b_{0}$ and $b_{i}$ $(i=1,3,\ldots ,N)$ are some constants,

$$\begin{aligned} \psi _{0}=\frac{\partial U_{0,\mu}}{\partial \mu}|_{\mu =1},\,\,\,\, \,\,\,\psi _{i}=\frac{\partial U_{0,1}}{\partial y_{i}},\,\,i=1,3, \ldots ,N. \end{aligned}$$

Proof

In view of $|\widetilde{\eta}_{m}|\leq C$, we may assume that $\widetilde{\eta}_{m}\rightarrow \eta$ in $C_{\mathrm{loc}}({\mathbb{R}}^{N})$. Then η satisfies

$$ -\Delta \eta =(2^{*}-1)U^{2^{*}-2}\eta ,\,\,\,x\in {\mathbb{R}}^{N}, $$

which gives $\eta =\sum \limits _{i=0}^{N}b_{i}\psi _{i}$. Since $\eta _{m}$ is even in $y_{2}$, we get $b_{2}=0$. □

We decompose

$$\begin{aligned} \eta _{m}(y)=&b_{0,m}\mu _{k_{m}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}}+b_{1,m} \mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} \\ &+\sum \limits _{i=3}^{N}b_{i,m}\mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}}+ \eta _{m}^{*}, \end{aligned}$$

(2.7)

where $\eta _{m}^{*}$ satisfies

$$\begin{aligned} \int _{{\mathbb{R}}^{N}}Z_{x_{{k_{m}},j},\mu _{k_{m}}}^{2^{*}-2} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}} \eta _{m}^{*}&=\int _{{\mathbb{R}}^{N}}Z_{x_{{k_{m}},j},\mu _{k_{m}}}^{2^{*}-2} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} \eta _{m}^{*} \\ &=\int _{{\mathbb{R}}^{N}}Z_{x_{{k_{m}},j},\mu _{k_{m}}}^{2^{*}-2} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}} \eta _{m}^{*}=0\,\,(i=3,\dots ,N). \end{aligned}$$

(2.8)

From Lemma 2.2 it follows that $b_{0,m}$, $b_{1,m}$, and $b_{i,m}$ $(i=3,\ldots,N)$ are bounded.

Lemma 2.3

We have

$$ \|\eta _{m}^{*}\|_{*}\leq C \mu _{k_{m}}^{-1-\epsilon}, $$

where $\epsilon >0$ is a small constant.

Proof

It is easy to see that

$$\begin{aligned} L_{k_{m}}\eta _{m}^{*}&=-\Delta \eta _{m}^{*}-(2^{*}-1)Q(|y'|,y'')u_{k_{m}}^{2^{*}-2} \eta _{m}^{*} \\ =&-(2^{*}-1)(Q(|y'|,y'')-1)\Big(b_{0,m}\mu _{k_{m}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}}+b_{1,m} \mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} \\ &\quad \quad \quad +\sum \limits _{i=3}^{N}b_{i,m}\mu _{k_{m}}^{-1} \sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}} \Big) \\ &\quad +(2^{*}-1)\sum \limits _{j=1}^{k_{m}}(u_{k_{m}}^{2^{*}-2}-U_{x_{{k_{m}},j}, \mu _{k_{m}}}^{2^{*}-2})\Big(b_{0,m}\mu _{k_{m}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}} \\ &\quad \quad \quad+b_{1,m} \mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} +\sum \limits _{i=3}^{N}b_{i,m}\mu _{k_{m}}^{-1} \sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}} \Big) \\ &\quad +2\nabla \eta \Big(b_{0,m}\mu _{k_{m}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}}+b_{1,m} \mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} \\ &\quad \quad \quad+ \sum \limits _{i=3}^{N}b_{i,m}\mu _{k_{m}}^{-1}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}} \Big) \\ &\quad +\Delta \eta \Big(b_{0,m}\mu _{k_{m}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \mu _{k_{m}}}+b_{1,m} \frac{1}{\mu _{k_{m}}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \overline{r}} \\ &\quad \quad \quad+ \sum \limits _{i=3}^{N}b_{i,m}\frac{1}{\mu _{k_{m}}}\sum \limits _{j=1}^{k_{m}} \frac{\partial Z_{x_{{k_{m}},j},\mu _{k_{m}}}}{\partial \bar{y}_{i}} \Big) \\ :&=I_{1}+I_{2}+I_{3}+I_{4}. \end{aligned}$$

Similarly to the proof of Lemma 2.4 in [15], we can prove that

$$\begin{aligned} \bigl\| I_{1}\bigr\| _{**}\leq C\mu _{k_{m}}^{-1-\epsilon},\,\,\,\,\,\, \,\,\,\,\,\,\bigl\| I_{2}\bigr\| _{**}\leq C\mu _{k_{m}}^{-1-\epsilon}. \end{aligned}$$

(2.9)

Similar to $J_{2}$ of Lemma 2.4 in [15], we can check that

$$\begin{aligned} |I_{3}| &\leq C\sum \limits _{j=1}^{k_{m}} \frac{\mu _{k_{m}}^{\frac{N}{2}}|\nabla \eta |}{(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{N-1}} \leq C\mu _{k_{m}}^{\frac{N+2}{2}}\sum \limits _{j=1}^{k_{m}} \frac{|\nabla \eta |}{\mu _{k_{m}}(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{N-1}} \\ &\leq C(\frac{1}{\mu _{k_{m}}})^{1+\epsilon}\sum \limits _{j=1}^{k_{m}} \frac{|\nabla \eta |\mu _{k_{m}}^{\frac{N+2}{2}}}{\mu _{k_{m}}^{-\epsilon}(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{N-1}} \\ &\leq C(\frac{1}{\mu _{k_{m}}})^{1+\epsilon} \sum \limits _{j=1}^{k_{m}} \frac{\mu _{k_{m}}^{\frac{N+2}{2}}}{(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{\frac{N+2}{2}+\tau}}. \end{aligned}$$

Hence $\|I_{3}\|_{**}\leq C(\frac{1}{\mu _{k_{m}}})^{1+\epsilon}$.

Also, similarly to $J_{3}$ of Lemma 2.4 in [15], we can prove that

$$\begin{aligned} |I_{4}|\leq & C\sum \limits _{j=1}^{k_{m}} \frac{\mu _{k_{m}}^{\frac{N-2}{2}}|\Delta \eta |}{(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{N-2}} \leq C\mu _{k_{m}}^{\frac{N+2}{2}}\sum \limits _{j=1}^{k_{m}} \frac{|\Delta \eta |}{\mu _{k_{m}}(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{N-1}} \\ \leq &C(\frac{1}{\mu _{k_{m}}})^{1+\epsilon} \sum \limits _{j=1}^{m} \frac{\mu _{k_{m}}^{\frac{N+2}{2}}}{(1+\mu _{k_{m}}|y-x_{{k_{m}},j}|)^{\frac{N+2}{2}+\tau}}, \end{aligned}$$

which yields that $\|I_{4}\|_{**}\leq C\Big(\displaystyle \frac{1}{\mu _{k_{m}}}\Big)^{1+ \epsilon}$.

It follows from (2.9) and $\|I_{3}\|_{**}$ and $\|I_{4}\|_{**}$ that

$$\begin{aligned} \|L_{k_{m}}\eta _{m}^{*}\|_{**}\leq C\mu _{k_{m}}^{-1-\epsilon}. \end{aligned}$$

Moreover, from (2.8) and Lemma 2.1 it follows that there exists $\rho >0$ such that

$$\begin{aligned} \|L_{k_{m}}\eta _{m}^{*}\|_{**}\geq \rho \|\eta _{m}^{*}\|_{*}. \end{aligned}$$

Thus the results follows. □

Lemma 2.4

If $(\tilde{Q})$ holds, then

$$ \tilde{\eta}_{m}\rightarrow 0 $$

uniformly in $C^{1}(B_{R}(0))$ for any $R>0$

Proof

Step 1. Recall that

$$\begin{aligned} \Omega _{j}=\Bigl\{ y=(y',y'')\in {\mathbb{R}}^{2}\times {\mathbb{R}}^{N-2}: \langle \frac{y{'}}{|y{'}|},\frac{x_{k,j}'}{|x_{k,j}'|}\rangle \geq \cos \frac{\pi}{k}\Bigr\} . \end{aligned}$$

To prove that $b_{i,k}\rightarrow 0$ $(i=1,3,\ldots ,N)$ in (2.7), we apply identities (2.3) in the domain $\Omega _{i}$ $(i=1,3,\ldots ,N)$:

$$\begin{aligned} &-\int _{\partial \Omega _{i}}\frac{\partial u_{k_{m}}}{\partial \nu} \frac{\partial \eta _{m}}{\partial y_{i}}-\int _{\partial \Omega _{i}} \frac{\partial \eta _{m}}{\partial \nu} \frac{\partial u_{k_{m}}}{\partial y_{i}} +\int _{\partial \Omega _{i}} \langle \nabla u_{k_{m}},\nabla \eta _{m}\rangle \nu _{i} +\int _{ \partial \Omega _{i}}Qu_{k_{m}}^{2^{*}-1}\eta _{m}\nu _{i} \\ &-\int _{\partial \Omega _{i}}u^{2^{*}-1}\eta _{m}\nu _{i} =-\int _{ \Omega _{i}}\frac{\partial Q}{\partial y_{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}. \end{aligned}$$

(2.10)

By the symmetry, $\frac{\partial u_{k_{m}}}{\partial \nu}=0$ and $\frac{\partial \eta _{m}}{\partial \nu}=0$ on $\partial \Omega _{i}$. So

$$\begin{aligned} &\text{the left-hand side of}~ \text{(2.10)} \\ &=\int _{\partial \Omega _{i}}\langle \nabla u_{k_{m}},\nabla \eta _{m} \rangle \nu _{i}-\int _{\partial \Omega _{i}}Q(|y'|,y'')u_{k_{m}}^{2^{*}-1} \eta _{m}\nu _{i} \\ &=-\sin \frac{\pi}{k_{m}}\Big(\int _{\partial \Omega _{i}}\langle \nabla u_{k_{m}},\nabla \eta _{m}\rangle -\int _{\partial \Omega _{i}}Q(|y'|,y'')u_{k_{m}}^{2^{*}-1} \eta _{m}\Big). \end{aligned}$$

(2.11)

Combining (2.10) and (2.11), we obtain $\langle \nu ,x_{k_{m},1}\rangle =-\sin \frac{\pi}{k_{m}}$, and thus

$$\begin{aligned} -\sin \frac{\pi}{k_{m}}\Big(\int _{\partial \Omega _{i}}\langle \nabla u_{k_{m}},\nabla \eta _{m}\rangle -\int _{\partial \Omega _{i}}Q(|y'|,y'')u_{k_{m}}^{2^{*}-1} \eta _{m}\Big) =\int _{\Omega _{i}}\frac{\partial Q}{\partial y_{i}}u_{k_{m}}^{2^{*}-1} \eta _{m}. \end{aligned}$$

(2.12)

To estimate the left-hand side in (2.12), we use (2.4) in $\Omega _{i}$. Applying the symmetry, we have

$$\begin{aligned} &\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q,y-x_{k_{m},i} \rangle =\int _{\partial \Omega _{i}}Q(|y'|,y'')u_{k_{m}}^{2^{*}-1} \eta _{m}\langle \nu ,y-x_{k_{m},i}\rangle \\ &-\int _{\partial \Omega _{i}}\langle \nabla u_{k_{m}},\nabla \eta _{m} \rangle \langle \nu ,y-x_{k_{m},i}\rangle . \end{aligned}$$

(2.13)

Since $\langle \nu ,y\rangle =0$ and $\langle \nu ,x_{k_{m},i}\rangle =-\sin \frac{i\pi}{k_{m}}$ on $\partial \Omega _{i}$, (2.13) becomes

$$\begin{aligned} \int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q,y-x_{k_{m},i} \rangle =&\sin \frac{\pi}{k_{m}}\Big(\int _{\partial \Omega _{i}}Q(|y'|,y'')u_{k_{m}}^{2^{*}-1} \eta _{m}-\int _{\partial \Omega _{i}}\langle \nabla u_{k_{m}}, \nabla \eta _{m}\rangle \Big). \end{aligned}$$

(2.14)

Combining (2.13) and (2.14), we obtain

$$\begin{aligned} \displaystyle \int _{\Omega _{i}}\frac{\partial Q}{\partial y_{i}}u_{k_{m}}^{2^{*}-1} \eta _{m}=\displaystyle \int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m} \langle \nabla Q,y-x_{k_{m},i}\rangle . \end{aligned}$$

(2.15)

Since $\nabla Q(x_{m_{k},i})=O(|x_{m_{k},i}-y_{0}|)=O(\mu _{k_{m}}^{-1- \epsilon})$ and

$$\begin{aligned} &\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}=\int _{(\Omega _{i})_{x_{k_{m}}, \mu _{k_{m}}}} \big(\mu _{k_{m}}^{-\frac{N-2}{2}}u_{k_{m}}(\mu _{k_{m}}^{-1}y+x_{k_{m},i}) \big)^{2^{*}-1}\tilde{\eta}_{m} \\ &=\int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\big[b_{0,m}\psi _{0}+b_{1,m} \psi _{1}+\sum \limits _{i=3}^{N}b_{i,m}\psi _{i} +\mu _{k_{m}}^{- \frac{N-2}{2}}\eta _{m}^{*}(\mu _{k_{m}}^{-1}y+x_{k_{m},i})\big]+O \big(\mu _{k_{m}}^{-1-\epsilon}\big) \\ &=O\big(\mu _{k_{m}}^{-1-\epsilon}\big), \end{aligned}$$

(2.16)

where $(\Omega _{i})_{x_{k_{m}},i}=\{y, \mu _{k_{m}}^{-1}y+x_{k_{m},i}\in \Omega _{i}\}$, we find

$$\begin{aligned} &\int _{\Omega _{i}}\frac{\partial Q}{\partial y_{i}}u_{k_{m}}^{2^{*}-1} \eta _{m}= \int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\Big( \frac{\partial Q}{\partial y_{i}}- \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}}\Big) +\int _{\Omega _{i}} \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}}u_{k_{m}}^{2^{*}-1} \eta _{m} \\ =&\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\Big[\langle \nabla \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}},y-x_{k_{m},i}\rangle + \frac{1}{2}\langle \nabla ^{2} \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}}(y-x_{k_{m},i}),y-x_{k_{m},i} \rangle \\ &+O(|y-x_{k_{m},i}|^{3})\Big]+O(\mu _{k_{m}}^{-2-\varepsilon}) \\ =&\int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\Big(b_{0,m}\psi _{0}+b_{1,m} \psi _{1}+\sum \limits _{l=3}^{N}b_{l,m}\psi _{l}\Big)\Big(\langle \nabla \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}}, \frac{y}{\mu _{k_{m}}}\rangle \\ &+\frac{1}{2}\langle \nabla ^{2} \frac{\partial Q(x_{k_{m},i})}{\partial y_{i}}\frac{y}{\mu _{k_{m}}}, \frac{y}{\mu _{k_{m}}}\rangle \Big)+O(\mu _{k_{m}}^{-2-\varepsilon}) \\ =& \frac{\frac{\partial ^{2} Q}{\partial y_{1}\partial y_{i}}(x_{m_{k},i})}{\mu _{k_{m}}}b_{1,m} \int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \psi _{1}y_{1}+\sum \limits _{l=3}^{N}b_{l,m} \frac{\frac{\partial ^{2} Q}{\partial y_{l}\partial y_{i}}(x_{k_{m},i})}{\mu _{k_{m}}} \int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\psi _{l}y_{l} \\ &+ \frac{\frac{\partial \triangle Q(x_{m_{k},i})}{\partial y_{i}}b_{0,m}}{2N\mu _{k_{m}}^{2}} \int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\psi _{0}|y|^{2} +O\big(\mu _{k_{m}}^{-2- \varepsilon}). \end{aligned}$$

(2.17)

Moreover, we can estimate

$$\begin{aligned} &\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q,y-x_{k_{m},i} \rangle \\ &=\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(y)- \nabla Q(x_{k_{m},i}),y-x_{k_{m},i}\rangle +\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1} \eta _{m}\langle \nabla Q(x_{k_{m},i}),y-x_{k_{m},i}\rangle \\ &=\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(y)- \nabla Q(x_{k_{m},i}),y-x_{k_{m},i}\rangle +O(\mu _{k_{m}}^{-2- \epsilon}) \\ &=\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla ^{2}Q(x_{k_{m},i})(y-x_{k_{m},i}),y-x_{k_{m},i} \rangle +O(\mu _{k_{m}}^{-2-\epsilon}) \\ &=\int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\Big(b_{0,m}\psi _{0}+b_{1,m} \psi _{1}+\sum \limits _{l=3}^{N}b_{l,m}\psi _{l}\Big) \big\langle \nabla ^{2}Q(x_{k_{m},i})\mu _{k_{m}}^{-1}y,\mu _{k_{m}}^{-1}y \big\rangle +O(\mu _{k_{m}}^{-2-\epsilon}) \\ &=\frac{b_{0,m}\Delta Q(x_{k_{m},i})}{N\mu _{k_{m}}^{2}}\int _{{ \mathbb{R}}^{N}}U^{2^{*}-1}\psi _{0}|y|^{2}+O\big(\mu _{k_{m}}^{-2- \epsilon}\big). \end{aligned}$$

(2.18)

Hence (2.17) and (2.18) give

$$\begin{aligned} &b_{0,m}\frac{1}{\mu _{k_{m}}}\Big( \frac{\frac{\partial \Delta Q}{\partial y_{i}}(x_{k_{m},i})}{2N} - \frac{\Delta Q(x_{k_{m},i})}{N}\Big)\displaystyle \int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \psi _{0}|y|^{2} \\ &+b_{1,m}\frac{\partial ^{2}Q}{\partial y_{1}\partial y_{i}}(x_{k_{m},i}) \int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \psi _{1}y_{1} +\sum \limits _{l=3}^{N}b_{l,m} \frac{\partial ^{2}Q}{\partial y_{l}\partial y_{i}}(x_{k_{m},i})\int _{{ \mathbb{R}}^{N}}U^{2^{*}-1}\psi _{l}y_{l} \\ =&O\big(\mu _{k_{m}}^{-1-\epsilon}\big). \end{aligned}$$

(2.19)

Step 2. Next, we apply (2.4) to obtain $\displaystyle \int _{{\mathbb{R}}^{N}}u_{k_{m}}^{2^{*}-1}\eta _{m} \langle \nabla Q(y),y\rangle =0$, which implies

$$ \displaystyle \int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(y),y\rangle =0. $$

(2.20)

On the other hand, proceeding as in the proof of (2.16), we find

$$\begin{aligned} &\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(x_{k_{m},i}),y \rangle \\ &=\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(x_{k_{m},i}),y-x_{k_{m},i} \rangle +\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(x_{k_{m},i}),x_{k_{m},i}\rangle \\ &=O\big(\mu _{k_{m}}^{-2-\epsilon}\big). \end{aligned}$$

Therefore from (2.19) we have

$$\begin{aligned} &\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(y),y \rangle \\ =&\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla Q(y)- \nabla Q(x_{k_{m},i}),y\rangle +O\big(\mu _{k_{m}}^{-2-\epsilon}\big) \\ =&\int _{\Omega _{i}}u_{k_{m}}^{2^{*}-1}\eta _{m}\langle \nabla ^{2}Q(x_{k_{m},i})(y-x_{k_{m},i}),y \rangle +O\big(\mu _{k_{m}}^{-2-\epsilon}\big) \\ =&\int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \Big(b_{0,m}\psi _{0}+b_{1,m} \psi _{1}+\sum \limits _{l=3}^{N}b_{l,m}\psi _{l}\Big)\langle \nabla ^{2}Q(x_{k_{m},i}) \mu _{k_{m}}^{-1}y, \mu _{k_{m}}^{-1}y+x_{k_{m},i}\rangle +O\big(\mu _{k_{m}}^{-2- \epsilon}\big) \\ =&\frac{b_{0,m}\Delta Q(x_{k_{m},i})}{N\mu _{k_{m}}^{2}}\int _{{ \mathbb{R}}^{N}}U^{2^{*}-1} \psi _{0}|y|^{2} + \frac{b_{1,m}}{\mu _{k_{m}}}\sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2} Q}{\partial y_{j}\partial y_{1}}(x_{m_{k},i}) \int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\psi _{1}y_{1} \\ &+ \frac{\sum \limits _{l=3}^{N}b_{l,m}\displaystyle \sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2} Q}{\partial y_{j}\partial y_{l}}(x_{k_{m},i})}{\mu _{k_{m}}} \int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\psi _{l}y_{l} +O\big(\mu _{k_{m}}^{-2- \epsilon}\big), \end{aligned}$$

which, combined with (2.20), implies that

$$\begin{aligned} &b_{0,m}\frac{\Delta Q(x_{k_{m},i})}{N\mu _{k_{m}}}\int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \psi _{0}|y|^{2} + b_{1,m}\sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2}Q}{\partial y_{j}\partial y_{1}}(x_{k_{m},i})\int _{{ \mathbb{R}}^{N}}U^{2^{*}-1} \psi _{1}y_{1} \\ & +\sum \limits _{l=3}^{N}b_{l,m}\sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2}Q}{\partial y_{j}\partial y_{l}}(x_{k_{m},i})\int _{{ \mathbb{R}}^{N}}U^{2^{*}-1}\psi _{l}y_{l} =O\big(\mu _{k_{m}}^{-1- \epsilon}\big). \end{aligned}$$

(2.21)

From (2.19) and (2.21) it follows that

$$\begin{aligned} &b_{1,m} \Big[\frac{\partial ^{2}Q}{\partial y_{1}\partial y_{i}}(x_{k_{m},i})- \big( \frac{\frac{\partial \Delta Q}{\partial y_{i}}(x_{k_{m},i})}{2\Delta Q(x_{k_{m},i})} -1\big) \sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2}Q}{\partial y_{j}\partial y_{1}}(x_{k_{m},i}) \Big]\int _{{\mathbb{R}}^{N}}U^{2^{*}-1} \psi _{1}y_{1} \\ &+\sum \limits _{l=3}^{N}b_{l,m} \Big[ \frac{\partial ^{2}Q}{\partial y_{l}\partial y_{i}}(x_{k_{m},i}) - \big( \frac{\frac{\partial \Delta Q}{\partial y_{i}}(x_{k_{m},i})}{2\Delta Q(x_{k_{m},i})} -1) \sum _{j=1}^{N}(x_{k_{m},i})_{j} \frac{\partial ^{2}Q}{\partial y_{j}\partial y_{l}}(x_{k_{m},i}) \Big] \int _{{\mathbb{R}}^{N}}U^{2^{*}-1}\psi _{l}y_{l} \\ & =O\big(\mu _{k_{m}}^{-1-\epsilon}\big)(i=1,3,4,\ldots ,N). \end{aligned}$$

(2.22)

By assumption $(\tilde{Q})$ we know that $b_{1,k}=o(1)$, $b_{l,k}=o(1)$ $(l=3,\ldots,n)$. Therefore $b_{0,k}=o(1)$. □

Now we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1

First, we have

$$\begin{aligned} |\eta _{m}(y)| &\leq C\int _{{\mathbb{R}}^{N}}\frac{1}{|z-y|^{N-2}}Q(z)|u^{2^{*}-2}_{k_{m}}(z)|| \eta _{m}(z)|dz \\ &\leq C\|\eta _{m}\|_{*}\int \frac{1}{|y-z|^{N-2}}|u^{2^{*}-2}_{k_{m}}(z)| \sum _{j=1}^{k_{m}} \frac{\eta _{m}^{\frac{N-2}{2}}}{(1+\eta _{m}|z-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau}} \\ &\leq C\|\eta _{m}\|_{*}\sum _{j=1}^{k_{m}} \frac{\eta _{m}^{\frac{N-2}{2}}}{(1+\eta _{m}|y-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau +\theta}} \end{aligned}$$

(2.23)

for some $\theta >0$.

Hence

$$ \frac{|\eta _{m}(y)|}{\displaystyle \sum _{j=1}^{k_{m}}\frac{\mu _{k_{m}}^{\frac{N-2}{2}}}{(1+\mu _{k_{m}}|y-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau}}} \leq C\|\eta _{m}\|_{*} \frac{\displaystyle \sum _{j=1}^{k_{m}}\frac{\mu _{k_{m}}^{\frac{N-2}{2}}}{(1+\mu _{k_{m}}|y-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau +\theta}}}{\displaystyle \sum _{j=1}^{k_{m}}\frac{\mu _{k_{m}}^{\frac{N-2}{2}}}{(1+\mu _{k_{m}}|y-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau}}}. $$

Since $\eta _{m}\rightarrow 0$ in $B_{R\mu ^{-1}_{k_{m}}}(x_{k_{m},j})$ and $\|\eta _{m}\|_{*}=1$, we know that

$$ \frac{|\eta _{m}(y)|}{\displaystyle \sum _{j=1}^{k_{m}}\frac{{\mu _{k_{m}}^{\frac{N-2}{2}}}}{(1+\mu _{k_{m}}|y-x_{k_{m},j}|)^{\frac{N-2}{2}+\tau}}} $$

attains its maximum in ${\mathbb{R}}^{N}\backslash \cup _{j=1}^{k_{m}}B_{R\mu ^{-1}_{k_{m}}}(x_{k_{m},j})$. Therefore $\|\eta _{m}\|_{*}\leq o(1)\|\eta _{m}\|_{*}$.

Hence $\|\eta _{m}\|_{*}\rightarrow 0$ as $m\rightarrow \infty$. This contradicts with $\|\eta _{m}\|_{*}=1$. □