Classification, non-degeneracy and existence of solutions to nonlinear Choquard equations

Abstract

In this paper, we first study the classification problem of positive solutions to the nonlocal Choquard equation

$$\begin{aligned} (-\Delta )^{\alpha }u=\left( \textrm{I}_{\beta }*\left| u\right| ^{p}\right) \left| u\right| ^{p-2}u, \qquad u\in {\mathcal {D}}^{\alpha ,2}\left( {\mathbb {R}}^N\right) \cap L^{\frac{2Np}{N+\beta }}\left( {\mathbb {R}}^N\right) , \end{aligned}$$

where $\alpha \in \left( 0,\frac{N}{2}\right)$ with $N\ge 1$, $\textrm{I}_{\beta }(x)$ is the standard Riesz potential with $\beta \in \left( 0,N\right)$ if $N\le 4\alpha$, or $\beta \in \left[ N-4\alpha ,N\right)$ if $N>4\alpha$, and $p\in [2, 2^*_{\alpha ,\beta }]$ with $2^*_{\alpha ,\beta }:=\frac{N+\beta }{N-2\alpha }$ which denotes the corresponding Hardy–Littlewood–Sobolev critical exponent. By applying the method of moving planes, we prove that the above Choquard equation has no positive solution in the subcritical case $p\in [2, 2^*_{\alpha ,\beta })$, and any positive solution must be of the form

$$\begin{aligned} U_{\lambda , x_0}(x)=C_{\alpha ,\beta ,N}\left( \frac{\lambda }{\lambda ^2 +\left| x-x^0\right| ^2}\right) ^{\frac{N-2\alpha }{2}} \end{aligned}$$

in the critical case $p=2^*_{\alpha ,\beta }$, where $x^0\in {\mathbb {R}}^N$, $\lambda >0$ and the constant $C_{\alpha ,\beta ,N}>0$ only depends on $\alpha$, $\beta$ and N. Moreover, we prove that this solution $U_{\lambda , x^0}(x)$ is non-degenerate. Based on the non-degeneracy result, we study the existence of solutions to the following Choquard equation with potential:

$$\begin{aligned} -\Delta u+V(\left| x\right| )u=\left( \textrm{I}_{\beta }*u^{2_{1, \beta }^*}\right) u^{2_{1,\beta }^*-1},\qquad u\in {\mathcal {D}}^{1,2}({\mathbb {R}}^N), \end{aligned}$$

where $\beta \in [N-4,N-2)$, $N\ge 5$, and $V(\cdot )$ is a bounded non-negative function. We show that this Choquard equation possesses infinitely many non-radial solutions provided that $r^{2}V(r)$ has an isolated local maximum point, or isolated local minimum point $r_0>0$ satisfying $V(r_0)>0$.

Appendices

Appendices

1.1 Proofs of Proposition 2.1 and Lemma 3.1

Proof of Proposition 2.1

Since $u\in {\mathcal {D}}^{\alpha ,2}({\mathbb {R}}^N)\cap L^{\frac{2Np}{N+\beta }}({\mathbb {R}}^N)$, using Sobolev embedding theorem, we have that $u\in L^q({\mathbb {R}}^N)$ for any q satisfies $\frac{2Np}{N+\beta }\le q\le \frac{2N}{N-2\alpha }$. Thus, by the definition of ${\bar{u}}$ in Proposition 2.1, there holds that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N}\left| u(x)\right| ^{q}\textrm{d}x&=\int _{{\mathbb {R}}^N}\frac{1}{\left| y-x^0\right| ^{2N}} \left| u\left( \frac{y-x^0}{\left| y-x^0\right| ^2}+x^0\right) \right| ^{q}\textrm{d}y\\&=\int _{{\mathbb {R}}^N}\left[ \frac{\left| {\bar{u}}(y)\right| ^p}{\left| y-x^0\right| ^{p\{\frac{2N}{r} -N+2\alpha \}}}\right] ^{\frac{q}{p}}\textrm{d}y. \end{aligned} \end{aligned}$$

Thus, we can obtain (2.3) and (2.4) by taking $q=\frac{2Np}{N+\beta }$ and $q=\frac{2N}{N-2\alpha }$, respectively.

Using [6, Lemma A.2.1.], we have that

$$\begin{aligned} (-\Delta )^\alpha {\bar{u}}(x)&=\frac{1}{\left| x-x^0\right| ^{N+2\alpha }} \left( (-\Delta )^\alpha u\right) \left( \frac{x-x^0}{\left| x-x^0\right| ^2}+x^0\right) \\&=\frac{{\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+2\alpha -(p-1)(N-2\alpha )}}\left( \textrm{I}_\beta *u^p\right) \left( \frac{x-x^0}{\left| x-x^0\right| ^2}+x^0\right) \\&=\frac{A_\beta {\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+2\alpha -(p-1)(N-2\alpha )}}\int _{{\mathbb {R}}^N}\frac{u^p(y)}{\left| y-x^0-\frac{x-x^0}{\left| x-x^0\right| ^2}\right| ^{N-\beta }}\textrm{d}y\\&\quad \left( \text {using the identity that}\ \left| y\right| \left| x-\frac{y}{\left| y\right| ^2}\right| =\left| x\right| \left| \frac{x}{\left| x\right| ^2}-y\right| \right) \\&=\frac{A_\beta {\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+2\alpha -(p-1)(N-2s)}}\\&\int _{{\mathbb {R}}^N}\frac{\left| x-x^0\right| ^{N-\beta }u^p(y-x^0+x^0)}{\left| y-x^0\right| ^{N-\beta } \left| \frac{y-x^0}{\left| y-x^0\right| ^2}-x+x^0\right| ^{N-\beta }} \textrm{d}y\\&\quad \left( \text {using the transform}\ z-x^0=\frac{y-x^0}{\left| y-x^0\right| ^2}\right) \\&=\frac{A_\beta {\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+2\alpha -(p-1)(N-2\alpha )}}\\&\quad \quad \int _{{\mathbb {R}}^N}\frac{\left| z-x^0\right| ^{N-\beta }\left| x-x^0\right| ^{N-\beta }u^p(\frac{z-x^0}{\left| z-x^0\right| ^2}+x^0)}{\left| z-x\right| ^{N-\beta }\left| z-x^0\right| ^{2N}}\textrm{d}z\\&=\frac{A_\beta {\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+2\alpha -(p-1)(N-2\alpha )}}\int _{{\mathbb {R}}^N}\frac{\left| x-x^0\right| ^{N-\beta }u^p(\frac{z-x^0}{\left| z-x^0\right| ^2}+x^0)}{\left| z-x^0\right| ^{N+\beta }\left| z-x\right| ^{N-\beta }}\textrm{d}z\\&=\frac{A_\beta {\bar{u}}^{p-1}\left( x\right) }{\left| x-x^0\right| ^{N+\beta -p(N-2\alpha )}}\int _{{\mathbb {R}}^N}\frac{{\bar{u}}^p(z)}{\left| z-x^0\right| ^{N+\beta -p(N-2\alpha )}\left| z-x\right| ^{N-\beta }}\textrm{d}z. \end{aligned}$$

Combining above argument, we can obtain the desired results in Proposition 2.1. $\square$

Lemma 5.1

[16, Lemma A.1.] Let $\gamma \in (0,N)$ and $\sigma _1>0$. Then, there exists a uniform constant $C>0$, such that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\frac{1}{\left| z-y\right| ^{N-\gamma }(1+\left| z\right| )^{\gamma +\sigma _1}}\textrm{d}z\le \left\{ \begin{array}{lllll} \frac{C}{(1+\left| y\right| )^{\sigma _1}}, &{}\quad \text {if}\ \gamma +\sigma _1<N, \\ \frac{C\left[ 1+\ln (1+\left| y\right| )\right] }{(1+\left| y\right| )^{\sigma _1}}, &{}\quad \text {if}\ \gamma +\sigma _1=N, \\ \frac{C}{(1+\left| y\right| )^{N-\gamma }}, &{}\quad \text {if}\ \gamma +\sigma _1>N. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.1)

Proof of Lemma 3.1

Let $\phi \in L^\infty ({\mathbb {R}}^N)$ be a solution of Eq. (3.4). By Lemma 5.1 and definition of $Q(\phi )$ in (3.5), we have that

$$\begin{aligned} \left| {\mathcal {Q}}(\phi )\right|&\le \frac{C}{(1+\left| x\right| )^{\beta +4\alpha -N}}\int _{{\mathbb {R}}^N}\frac{1}{\left| x-y\right| ^{N-\beta } \left[ 1+\left| y\right| \right] ^{N+\beta }}\textrm{d}y\\&\quad + \frac{C}{(1+\left| x\right| )^{\beta +2\alpha }}\int _{{\mathbb {R}}^N} \frac{1}{\left| x-y\right| ^{N-\beta }\left[ 1+\left| y\right| \right] ^{2\alpha +\beta }}\textrm{d}y\\&\le \frac{C}{(1+\left| x\right| )^{4\alpha }}. \end{aligned}$$

Thus, by Eq. (3.4), we have that

$$\begin{aligned} \left| \phi (x)\right|= & {} C(N,\alpha )\left| \int _{{\mathbb {R}}^N}\frac{{\mathcal {Q}} (\phi )(y)}{\left| x-y\right| ^{N-2\alpha }}\textrm{d}y\right| \le \left\{ \begin{array}{lllll} \frac{C}{(1+\left| y\right| )^{2\alpha }}, &{}\quad \text {if}\ \alpha <\frac{N}{4}, \\ \frac{C\left[ 1+\ln (1+\left| y\right| )\right] }{(1+\left| y\right| )^{2\alpha }}, &{}\quad \text {if}\ \alpha =\frac{N}{4}, \\ \frac{C}{(1+\left| y\right| )^{N-2\alpha }}, &{}\quad \text {if}\ \alpha \in (\frac{N}{4},\frac{N}{2}). \end{array} \right. \nonumber \\ \end{aligned}$$

(5.2)

Therefore, we need to consider the case $\alpha \in (0,\frac{N}{4}]$. We will perform an iterative process. By the above estimate (5.2), we have that if $4\alpha \in (0,N)$

$$\begin{aligned} \left| {\mathcal {Q}}(\phi )\right|&\le \frac{C}{(1+\left| x\right| )^{\beta +6\alpha -N}}\int _{{\mathbb {R}}^N} \frac{1}{\left| x-y\right| ^{N-\beta }\left[ 1+\left| y\right| \right] ^{N+\beta }}\textrm{d}y\\&\quad +\frac{C}{(1+\left| x\right| )^{\beta +2\alpha }}\int _{{\mathbb {R}}^N} \frac{1}{\left| x-y\right| ^{N-\beta }\left[ 1+\left| y\right| \right] ^{4\alpha +\beta }}\textrm{d}y\\&\le \frac{C}{(1+\left| x\right| )^{6\alpha }}, \end{aligned}$$

and if $4\alpha =N$

$$\begin{aligned} \left| {\mathcal {Q}}(\phi )\right|&\le \frac{C\left[ 1+\ln (1+\left| y\right| ) \right] }{(1+\left| x\right| )^{\beta +6\alpha -N}}\int _{{\mathbb {R}}^N} \frac{1}{\left| x-y\right| ^{N-\beta }\left[ 1+\left| y\right| \right] ^{N+\beta }}\textrm{d}y\\&\quad +\frac{C}{(1+\left| x\right| )^{\beta +2\alpha }}\int _{{\mathbb {R}}^N} \frac{\left[ 1+\ln (1+\left| y\right| )\right] }{\left| x-y\right| ^{N-\beta } \left[ 1+\left| y\right| \right] ^{4\alpha +\beta }}\textrm{d}y\\&\le \frac{C}{(1+\left| x\right| )^{5\alpha }}. \end{aligned}$$

Thus, we have that

$$\begin{aligned} \left| \phi (x)\right| =C(N,\alpha )\left| \int _{{\mathbb {R}}^N}\frac{{\mathcal {Q}} (\phi )(y)}{\left| x-y\right| ^{N-2\alpha }}\textrm{d}y\right| \le \left\{ \begin{array}{lllll} \frac{C}{(1+\left| y\right| )^{4\alpha }}, &{}\quad \text {if}\ \alpha <\frac{N}{6}, \\ \frac{C\left[ 1+\ln (1+\left| y\right| )\right] }{(1+\left| y\right| )^{4\alpha }}, &{}\quad \text {if}\ \alpha =\frac{N}{6}, \\ \frac{C}{(1+\left| y\right| )^{N-2\alpha }}, &{}\quad \text {if}\ \alpha \in (\frac{N}{6},\frac{\alpha }{4}]. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.3)

According the above argument, for any $\alpha \in (\frac{N}{6},\frac{N}{2})$, there holds that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^N}\frac{{\mathcal {Q}}(\phi )(y)}{\left| x-y\right| ^{N-2\alpha }}\textrm{d}y\right| \le \frac{C}{(1+\left| y\right| )^{N-2\alpha }}. \end{aligned}$$

Thus, for any given $\alpha \in (0,\frac{N}{2})$, after finite times of iterations as shown above, we can get the desired estimate (3.6). $\square$

1.2 Some useful estimates and energy expansion

Lemma 5.2

Let $N\ge 5$, $\beta \in [N-4,N-2)$ and $\gamma \ge \frac{\min \{N-2,N-\beta \}-2}{\min \{N-2,N-\beta \}}$. Then, we have that for any $i=1,\ldots ,k$

$$\begin{aligned} \sum _{j=1,j\ne i}^k\frac{1}{\left| \Lambda (y^{j,k}-y^{i,k})\right| ^{\gamma }}\le \left\{ \begin{array}{lllll} Ck^{-\frac{2\gamma }{\min \{N-2,N-\beta \}-2}},&{}\quad \text {if}\ \ \gamma >1,\\ Ck^{-\frac{2}{\min \{N-2,N-\beta \}-2}}\ln k, &{}\quad \text {if}\ \ \gamma =1,\\ Ck^{1-\frac{(\min \{N-2,N-\beta \})\gamma }{\min \{N-2,N-\beta \}-2}}, &{}\quad \text {if}\ \ \frac{\min \{N-2,N-\beta \}-2}{\min \{N-2,N-\beta \}}\\ \le \gamma <1. \end{array} \right. \nonumber \\ \end{aligned}$$

(5.4)

where these points $y^{j,k}$ with $j=1,\ldots ,k$ are defined by (1.14), and $\Lambda \in \left[ L_0k^{\frac{N-2}{N-4}}, L_1k^{\frac{N-2}{N-4}}\right]$ for some constants $L_1>L_0>0$.

Proof

By definition (1.14), we get that

$$\begin{aligned}&\sum _{j=1,j\ne i}^k\frac{1}{\left| \Lambda (y^{j,k}-y^{i,k})\right| ^{\gamma }} =\sum _{j=1,j\ne i}^k\frac{1}{\left| 2\Lambda r\sin \frac{(j-i)\pi }{k}\right| ^{\gamma }} \le C\frac{1}{(r\Lambda )^\gamma }\sum _{j=1}^k\left( \frac{j}{k}\right) ^{-\gamma }\\&\le \left\{ \begin{array}{ll} Ck^{-\frac{2\gamma }{\min \{N-2,N-\beta \}-2}}&{}\quad \text {if}\ \gamma >1,\\ Ck^{-\frac{2}{\min \{N-2,N-\beta \}-2}}\left( 1+\int _1^{k} \xi ^{-1}\textrm{d}\xi \right) \\ \le Ck^{-\frac{2}{\min \{N-2,N-\beta \}-2}}\ln k &{}\quad \text {if}\ \gamma =1,\\ Ck^{-\frac{2\gamma }{\min \{N-2,N-\beta \}-2}}\left( 1 +\int _1^{k}\xi ^{-\gamma }\textrm{d}\xi \right) \\ \le Ck^{1-\frac{(\min \{N-2,N-\beta \})\gamma }{\min \{N-2,N-\beta \}-2}} &{}\quad \text {if}\ \ \frac{\min \{N-2,N-\beta \}-2}{\min \{N-2,N-\beta \}}\le \gamma <1. \end{array} \right. \end{aligned}$$

$\square$

Lemma 5.3

Let $\beta \in [N-4,N-2)$ and $N\ge 5$

$$\begin{aligned} {\mathbb {I}}(z):=\int _{{\mathbb {R}}^N}\frac{1}{\left| y-z\right| ^{N-\beta }} \left[ \sum _{j=1}^k\left( \frac{1}{1+\left| y-\Lambda y^{j,k}\right| ^2}\right) ^{\frac{N-2}{2}}\right] ^{\frac{N+\beta }{N-2}}\textrm{d}y. \end{aligned}$$

Then, there holds that

$$\begin{aligned} {\mathbb {I}}(z)\le C(\beta ,N)\sum _{i=1}^k\frac{1}{\left( 1+\left| z-\Lambda y^{i,k}\right| \right) ^{N-\beta }}, \end{aligned}$$

for some positive constant $C(\beta ,N)>0$ which only depends on $\beta$ and N.

Proof

Using the facts that for any $y\in \Omega _i$, $j=1,\ldots ,k$ and $j\ne i$, $\left| y-\Lambda y^{j,k}\right| \ge \left| y-\Lambda y^{i,k}\right|$ and

$$\begin{aligned} \left| y-\Lambda y^{j,k}\right| ^2&\ge \left| y_1-\Lambda r\cos \frac{2(j-1) \pi }{k}\right| ^2+\left| y_2-\Lambda r\sin \frac{2(j-1)\pi }{k}\right| ^2\nonumber \\&\quad \left( y_1={\tilde{r}}\cos \theta ,\ \ y_2={\tilde{r}}\sin \theta ,\ \ \theta \in \left( \frac{(2i-3)\pi }{k},\frac{(2i-1)\pi }{k}\right) \right) \nonumber \\&\ge \left( \Lambda r-{\tilde{r}}\right) ^2+4\Lambda r {\tilde{r}}\sin ^2\left( \frac{(j-1)\pi }{k}-\frac{\theta }{2}\right) \nonumber \\&\ge \frac{1}{4}(\Lambda r)^2\sin ^2\frac{(j-i)\pi }{k}=\frac{1}{16} \left| \Lambda y^{i,j}-\Lambda y^{j,k}\right| ^2, \end{aligned}$$

(5.5)

and Lemmas 5.1-5.2, we have that

$$\begin{aligned} {\mathbb {I}}(z)&=\sum _{i=1}^k\int _{\Omega _i}\frac{1}{\left| y-z\right| ^{N-\beta }} \left[ \sum _{j=1}^k\left( \frac{1}{1+\left| y-\Lambda y^{j,k}\right| ^2} \right) ^{\frac{N-2}{2}}\right] ^{\frac{N+\beta }{N-2}}\textrm{d}y\\&\le C(\beta ,N)\sum _{i=1}^k\int _{\Omega _i} \frac{1}{\left| y-z\right| ^{N-\beta }}\left[ \sum _{j=1,j\ne i}^k \left( \frac{1}{1+\left| y-\Lambda y^{j,k}\right| }\right) ^{N-2} \right] ^{\frac{N+\beta }{N-2}}\textrm{d}y\\&\quad +C(\beta ,N)\sum _{i=1}^k\int _{\Omega _i}\frac{1}{\left| y-z\right| ^{N -\beta }}\left( \frac{1}{1+\left| y-\Lambda y^{i,k}\right| }\right) ^{N+\beta }\textrm{d}y\\&\le C(\beta ,N)\sum _{i=1}^k\int _{\Omega _i}\frac{1}{\left| y-z\right| ^{N -\beta }}\left( \frac{1}{1+\left| y-\Lambda y^{i,k}\right| }\right) ^{N+\beta -\frac{(N+\beta )(N-4)}{(N-2)^2}}\\&\quad \times \left[ \sum _{j=1,j\ne i}^k\left( \frac{1}{\Lambda \left| y^{i,k}-y^{j,k}\right| }\right) ^{\frac{N-4}{N-2}}\right] ^{\frac{N+\beta }{N-2}}\textrm{d}y\\&\quad +C(\beta ,N)\sum _{i=1}^k\int _{\Omega _i}\frac{1}{\left| y-z\right| ^{N -\beta }}\left( \frac{1}{1+\left| y-\Lambda y^{i,k}\right| }\right) ^{N+\beta }\textrm{d}y\\&\le C(\beta ,N)\sum _{i=1}^k\frac{1}{\left( 1+\left| z-\Lambda y^{i,k}\right| \right) ^{N-\beta }}, \end{aligned}$$

where we notice that $\beta >\frac{(N+\beta )(N-4)}{(N-2)^2}$. $\square$

Lemma 5.4

(Energy expansion) Let $\beta \in [N-4,N-2)$ and $N\ge 5$. Then, there holds that

$$\begin{aligned} \textrm{E}(W_{\lambda _k,r_k})&= k\left\{ B_0+\frac{B_1V(r)}{\Lambda ^2}- \frac{B_2k^{-\min \{N-\beta ,N-2\}}}{(r\Lambda )^{\min \{N -\beta ,N-2\}}}+O\left( \frac{1}{\Lambda ^{2+\sigma }}\right) \right\} , \end{aligned}$$

where $B_0$, $B_1$ and $B_2$ are fixed positive constants, $\sigma >0$ is a small constant and $y^{i,k}$ is defined by (1.14) for $i=1,\ldots ,k$.

Proof

Recall that

$$\begin{aligned} \textrm{E}(W_{\Lambda ,r}){=}\frac{1}{2}\int _{{\mathbb {R}}^N}\left[ \left| \nabla W_{\Lambda ,r}\right| ^2{+}V(\left| z\right| )\left| W_{\Lambda ,r}\right| ^2\right] {-}\frac{1}{2\times 2^*_\beta }\int _{{\mathbb {R}}^N}\left[ \textrm{I}_{\beta }* W_{\Lambda ,r}^{2^*_\beta }\right] W_{\Lambda ,r}^{2^*_\beta },\nonumber \\ \end{aligned}$$

(5.6)

where $W_{\Lambda ,r}$ is defined by (1.13) and $2^*_\beta =\frac{N+\beta }{N-2}$. We next calculate each term in the above expansion.

By the definition of $W_{\Lambda ,r}$ in (1.13) and rotation transformation, we have that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\left| \nabla W_{\Lambda ,r}\right| ^2&=\sum _{i,j=1}^k\int _{{\mathbb {R}}^N} \left[ H^{2^*_\beta }_{\Lambda ,y^{j,k}}*\textrm{I}_{\beta }\right] H^{2^*_\beta -1}_{\Lambda ,y^{j,k}}H_{\Lambda ,y^{i,k}}\\&=k\left[ \int _{{\mathbb {R}}^N} \left[ H^{2^*_\beta }_{1,0}*\textrm{I}_{\beta } \right] H^{2^*_\beta }_{1,0}\right. \\&\left. +\sum _{i=2}^k\int _{{\mathbb {R}}^N} \left[ H^{2^*_\beta }_{\Lambda ,y^{1,k}}*\textrm{I}_{\beta } \right] H^{2^*_\beta -1}_{\Lambda ,y^{1,k}}H_{\Lambda ,y^{i,k}}\right] \\&=k\left[ \int _{{\mathbb {R}}^N} \left[ H^{2^*_\beta }_{1,0}*\textrm{I}_{\beta } \right] H^{2^*_\beta }_{1,0}+\sum _{i=2}^k\frac{C_0}{\Lambda ^{N-2} \left| y^{i,k}-y^{1,k}\right| ^{N-2}}\right. \\&\qquad \left. +O\left( \frac{k^{N-1}}{\Lambda ^{N-1}}\right) \right] , \end{aligned}$$

where we used Lemma 5.1 and the identity (see (37) in [8]) that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\frac{1}{\left| x-y\right| ^{2s}}\left( \frac{1}{1+\left| y\right| ^2} \right) ^{N-s}\textrm{d}y={\bar{C}}(N,s)\left( \frac{1}{1+\left| x\right| ^2}\right) ^s, \end{aligned}$$

(5.7)

for $2s\in (0,N)$ and some positive constant ${\bar{C}}(N,s)$.

Similarly, for the second term in (5.6), we have that

$$\begin{aligned}&\int _{{\mathbb {R}}^N}V(\left| z\right| )\left| W_{\Lambda ,r}\right| ^2\\&\quad =k\left[ \int _{{\mathbb {R}}^N} V(\left| z\right| )H^2_{\Lambda ,y^{1,k}}+\sum _{i=2}^k\int _{{\mathbb {R}}^N} V(\left| z\right| ) H_{\Lambda ,y^{1,k}}H_{\Lambda ,y^{i,k}}\right] \\&\quad =k\left[ \Lambda ^{-2}\int _{{\mathbb {R}}^N}V(\left| \Lambda ^{-1}z+y^{1,k}\right| ) H^2_{1,0}+\sum _{i=2}^k\int _{{\mathbb {R}}^N} O\left( H_{\Lambda ,y^{1,k}} H_{\Lambda ,y^{i,k}}\right) \right] \\&\quad =k\left[ \Lambda ^{-2}V(\left| y^{1,k}\right| )\int _{{\mathbb {R}}^N}H^2_{1,0} +O\left( \sum _{i=2}^k\frac{1}{\Lambda ^{N-2}\left| y^{1,k}-y^{i,k}\right| ^{N -4}}\right) \right. \\&\qquad \left. +O\left( \Lambda ^{-2-\frac{\theta }{2}}\right) \right] , \end{aligned}$$

where we used the fact that $V(\left| z\right| )\in \textrm{C}^{0,\theta }((r_0-\varepsilon ,r_0+\varepsilon ))$.

We need to consider the last term in (5.6). By the fact that $W_{\Lambda ,r} \in \Xi _s$ in (1.11), we get that

$$\begin{aligned}&\int _{{\mathbb {R}}^N}\left[ \textrm{I}_{\beta }*W_{\Lambda ,r}^{2^*_\beta } \right] W_{\Lambda ,r}^{2^*_\beta }\\&\quad =k\int _{\Omega _1} \left[ \textrm{I}_{\beta }*W_{\Lambda ,r}^{2^*_\beta } \right] W_{\Lambda ,r}^{2^*_\beta }\\&\quad =k\int _{\Omega _1}\left[ \textrm{I}_{\beta }*W_{\Lambda , r}^{2^*_\beta }\right] H_{\Lambda ,y^{1,k}}^{2^*_\beta } + k2^*_\beta \int _{\Omega _1}\left[ \textrm{I}_{\beta }* W_{\Lambda ,r}^{2^*_\beta }\right] H_{\Lambda ,y^{1,k} }^{2^*_\beta -1}\sum _{i=2}^kH_{\Lambda ,y^{i,k}}\\&\qquad {+}k\int _{\Omega _1}\left[ \textrm{I}_{\beta }*W_{\Lambda , r_k}^{2^*_\beta }\right] \left\{ O\left( H_{\Lambda ,y^{1,k} }^{2_\beta ^*{-}2}{+}\left[ \sum _{i=2}^kH_{\Lambda ,y^{i,k}} \right] ^{2_\beta ^*{-}2}\right) \left[ \sum _{i{=}2}^kH_{\Lambda ,y^{i,k}}\right] ^{2}\right\} \\&\quad =\tilde{{\mathfrak {J}}}_1+\tilde{{\mathfrak {J}}}_2+\tilde{{\mathfrak {J}}}_3, \end{aligned}$$

where the terms $\tilde{{\mathfrak {J}}}_1$, $\tilde{{\mathfrak {J}}}_2$ and $\tilde{{\mathfrak {J}}}_3$ are defined by the last equality.

We notice that for any $x\in \Omega _i$ with $i=1,\ldots ,k$

$$\begin{aligned} W_{\Lambda ,r}^{2^*_\beta }(x)&=H_{\Lambda ,y^{i,k}}^{2^*_\beta } +2^*_\beta H_{\Lambda ,y^{i,k}}^{2^*_\beta -1}\sum _{j=1,j\ne i}^kH_{\Lambda ,y^{j,k}}\\&\quad +O\left( H_{\Lambda ,y^{i,k}}^{2_\beta ^*-2} +\left[ \sum _{j=1,j\ne i}^kH_{\Lambda ,y^{j,k}}\right] ^{2_\beta ^* -2}\right) \left[ \sum _{j=1,j\ne i}^kH_{\Lambda ,y^{j,k}}\right] ^{2}. \end{aligned}$$

Thus, by Lemmas 5.1-5.2, HLS inequality (1.5), and identity (5.7), we have that

$$\begin{aligned} \tilde{{\mathfrak {J}}}_1&=k\sum _{i=1}^k\int _{\Omega _1} \int _{\Omega _i}\frac{A_\beta }{\left| x-z\right| ^{N-\beta }}W_{\Lambda , r}^{2^*_\beta }(z)\textrm{d}zH^{2^*_\beta }_{\Lambda ,y^{1,k}}(y)\textrm{d}y\\&=k\int _{\Omega _1}\left[ \textrm{I}_{\beta }*H^{2^*_\beta }_{\Lambda ,y^{1,k}}\right] W_{\Lambda ,r}^{2^*_\beta } \\&\quad +k\sum _{i=2}^k \int _{\Omega _1}\int _{\Omega _i}\frac{A_\beta }{\left| x-z\right| ^{N -\beta }}H_{\Lambda ,y^{i,k}}^{2^*_\beta }(z)\textrm{d} zH^{2^*_\beta }_{\Lambda ,y^{1,k}}(y)\textrm{d}y\\&\quad +kO\left( \Vert W_{\Lambda ,r}^{2^*_\beta }\Vert _{L^{\frac{3N}{2(N+\beta )}}(\Omega _1)} \Vert H_{\Lambda ,y^{1,k}}^{2^*_\beta }\Vert _{L^{\frac{3N}{N+\beta }} ({\mathbb {R}}^N\setminus \Omega _1)}\right) +kO\left( k^{N}\Lambda ^{-N}\right) \\&=k\int _{\Omega _1}\left[ \textrm{I}_{\beta }*H^{2^*_\beta }_{\Lambda , y^{1,k}}\right] \left\{ H_{\Lambda ,y^{1,k}}^{2^*_\beta }+2^*_\beta H_{\Lambda ,y^{1,k}}^{2^*_\beta -1}\sum _{j=2}^kH_{\Lambda ,y^{j,k}}\right\} \\&\quad +k\sum _{i=2}^k\int _{\Omega _1}\int _{\Omega _i}\frac{A_\beta }{\left| x-z\right| ^{N-\beta }}H_{\Lambda ,y^{i,k}}^{2^*_\beta }(z)\textrm{d}z H^{2^*_\beta }_{\Lambda ,y^{1,k}}(y)\textrm{d}y\\&\quad +kO\left( k^\frac{2(N+\beta )}{3}\Lambda ^{-\frac{2(N +\beta )}{3}}\right) +kO\left( k^{N-1}\Lambda ^{-(N-1)}\right) \\&=k\left[ \int _{{\mathbb {R}}^N}\left[ \textrm{I}_{N-4}*H^{2^*_\beta }_{1,0} \right] H^{2^*_\beta }_{1,0}+\sum _{j=2}^k\frac{C_1\Lambda ^{\beta -N}}{\left| y^{j,k}-y^{1,k}\right| ^{N-\beta }}+\sum _{j=2}^k\frac{2^*_\beta C_0\Lambda ^{2-N}}{\left| y^{j,k}-y^{1,k}\right| ^{N-2}}\right. \\&\quad \left. +O\left( \frac{1}{\Lambda ^{2+\sigma }}\right) \right] , \end{aligned}$$

and

$$\begin{aligned} \tilde{{\mathfrak {J}}}_2&=k2^*_\beta \int _{\Omega _1}\left[ \textrm{I}_{\beta }* W_{\lambda _k,r_k}^{2^*_\beta }\right] H_{\lambda _k,y^{1,k}}^{2^*_\beta -1} \sum _{i=2}^kH_{\lambda _k,y^{i,k}}\\&=k2^*_\beta C(N,\beta )\int _{\Omega _1}\left( \frac{\lambda _k}{1 +\lambda _k^2\left| z-y^{i,k}\right| ^2}\right) ^\frac{N+2}{2} \sum _{i=2}^kH_{1,\lambda _ky^{i,k}}+kO\left( \frac{k^{N-1}}{\lambda _k^{N-1}}\right) \\&=k\left[ \sum _{i=2}^k\frac{2^*_\beta C_0}{\lambda _k^{N-2} \left| y^{i,k}-y^{1,k}\right| ^{N-2}}+O\left( \frac{1}{\lambda _k^{2+\sigma }}\right) \right] , \end{aligned}$$

where $\sigma >0$ is a small constant, $C_0$ and $C_1$ are fixed positive constants.

Similarly, we can get that

$$\begin{aligned} \tilde{{\mathfrak {J}}}_3&=k\int _{\Omega _1}\left[ \textrm{I}_{\beta }*W_{\Lambda ,r}^{2^*_\beta }\right] \left\{ O\left( H_{\Lambda ,y^{1,k}}^{2_\beta ^*-2}+\left[ \sum _{i=2}^kH_{\Lambda ,y^{i,k}}\right] ^{2_\beta ^*-2}\right) \left[ \sum _{i=2}^kH_{\Lambda ,y^{i,k}}\right] ^{2}\right\} \\&\le Ck\int _{\Omega _1}H_{\Lambda ,y^{1,k}}^{\frac{N+1}{N-2}}\left[ \sum _{i=2}^kH^\frac{N-1}{2(N-2)}_{\Lambda ,y^{i,k}}\right] ^{2}\\&\quad +Ck\int _{\Omega _1}H_{\Lambda ,y^{1,k}}^{\frac{N+1/2}{N-2}}\left[ \sum _{i=2}^kH^\frac{1}{2}_{\Lambda ,y^{i,k}}\right] ^{2}\left\{ \sum _{i=2}^kH^\frac{3}{2(N-2)}_{\Lambda ,y^{i,k}}\right\} \\&\quad +Ck\int _{\Omega _1}H_{\Lambda ,y^{1,k}}^{\frac{N+1}{N-2}}\left[ \sum _{i=2}^kH^\frac{N-1}{N+\beta }_{\Lambda ,y^{i,k}}\right] ^{2_\beta ^*}\\&\quad +Ck\int _{\Omega _1}H_{\Lambda ,y^{1,k}}^{\frac{N+1/2}{N-2}}\left[ \sum _{i=2}^kH^\frac{N-2}{N+\beta }_{\Lambda ,y^{i,k}}\right] ^{2_\beta ^*}\left\{ \sum _{i=2}^kH^\frac{3}{2(N-2)}_{\Lambda ,y^{i,k}}\right\} \\&\le C\frac{k^{N-1}}{\Lambda ^{N-1}}. \end{aligned}$$

Combining the above arguments and the proof of Lemma 5.2, we can obtain the desired result. $\square$

Moreover, according to a similar argument in above proof and the fact that

$$\begin{aligned} \frac{\partial H_{\Lambda ,y}}{\partial \Lambda }(z)= \frac{N-2}{2\Lambda }\left( \frac{1-\Lambda ^2\left| z-y\right| ^2}{1+\Lambda ^2\left| z-y\right| ^2}\right) H_{\Lambda ,y}(z), \end{aligned}$$

we can get the following asymptotic expansion of $\frac{\partial \textrm{E}(W_{\Lambda ,r})}{\partial \Lambda }$.

Lemma 5.5

There holds that

$$\begin{aligned}&\frac{\partial \textrm{E}(W_{\Lambda ,r})}{\partial \Lambda }\\&=k\left\{ -\frac{2B_1V(r_k)}{\Lambda ^3}+ \frac{\min \{N-\beta ,N-2\}B_2k^{\min \{N-\beta ,N-2\}}}{\Lambda ^{\min \{N-\beta +1,N-1\}}r^{\min \{N-\beta ,N-2\}}} +O\left( \frac{1}{\Lambda ^{3+\sigma }}\right) \right\} , \end{aligned}$$

where $B_1$, $B_2$ and $\sigma >0$ are given by Lemma 5.4, $y^{i,k}$ is defined by (1.14) for $i=1,\ldots ,k$ and $r\in [r_0-\delta ,r_0+\delta ]$.