Non-radial solutions for the fractional Hénon equation with critical exponent

Abstract

We consider the following fractional Hénon type equation with critical growth:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=K(|y|)u^{\frac{N+2s}{N-2s}}, \hbox { }u>0, &{}\hbox { }y\in B_1(0), \\ \displaystyle u=0,&{}\hbox { } y\in B_{1}^{c}(0), \end{array}\right. } \end{aligned}$$

(0.1)

where K(|y|) is a bounded function defined in [0, 1], $B_1(0)$ is the unit ball in ${\mathbb {R}}^{N}$, $N\ge 3$ for $\frac{3}{4} \le s<1$ and $3\le N < 2s -1 +\frac{2}{3-4s}$ for $\frac{11-\sqrt{41}}{8}< s<\frac{3}{4}$. We show that if $K(1)>0$ and $K'(1)>0$, then equation (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. The most ingredients of the paper are using the Green representation and estimating the Green function and its regular part very carefully. For this purposes, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

Appendices

Appendix A. The estimate of the projection

In both appendices, we always assume that

$$\begin{aligned} x_j=\left( r\cos \frac{2(j-1)\pi }{k},r\sin \frac{2(j-1)\pi }{k},0\right) ,\hbox { }j=1,...,k, \end{aligned}$$

where 0 is the zero vector in ${\mathbb {R}}^{N-2}$, and $r\in [ \mu (1-\frac{r_0}{k}), \mu (1-\frac{r_1}{k}) ]$. Recall that ${\bar{x}}_j=\frac{1}{\mu }x_j.$ G(x, y) be the Green function of $(-\Delta )^{s}$ in $B_{1}(0)$ with homogenous Dirichlet boundary condition. H(x, y) be the part of the Green function. $PU_{x,\Lambda }$ is the solution of (1.4) and $r_3 =\min ( r_0,1)$.

For $l=1,..,N$, we denote $\partial _l PU_{x,\Lambda }(y) = \frac{\partial PU_{x,\Lambda } (y) }{\partial x_l}$, $\partial _l U_{x,\Lambda }(y) = \frac{\partial U_{x,\Lambda } (y) }{\partial x_l}$, and $\partial _l H(x,y) = \frac{\partial H }{\partial x_l}(x,y)$. For $l=N+1$, we set $\partial _l PU_{x,\Lambda }(y) = \frac{\partial PU_{x,\Lambda } (y) }{\partial \Lambda }$, and $\partial _l U_{x,\Lambda }(y) = \frac{\partial U_{x,\Lambda } (y) }{\partial \Lambda }$.

Lemma A.1

Assume $N\ge 3$. For any $i=1,...,k$, if $y\in B_{{\frac{\mu r_3}{8k}}}(x_j)$ and $j\ne i$, then

$$\begin{aligned}&U_{x_i,\Lambda }(y) - PU_{x_i,\Lambda }(y) \nonumber \\&\quad = \frac{A_{N,s}H(\bar{x_i},\mu ^{-1}y)}{ \Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s}} + O\left( \frac{1}{\mu ^{N}|\bar{x_i} -\mu ^{-1} y |^{N} } +\frac{k^{2s}}{\mu ^{N}|\bar{x_i} -\mu ^{-1}y |^{N-2s}}\right) , \end{aligned}$$

(A.1)

$$\begin{aligned}&\partial _l U_{x_i,\Lambda }(y) - \partial _l PU_{x_i,\Lambda }(y) \nonumber \\ {}&\quad = \frac{A_{N,s}\partial _l H(\bar{x_i},\mu ^{-1}y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s+1}} + O\left( \frac{1}{\mu ^{N+1}|\bar{x_i} -\mu ^{-1} y |^{N+1} } +\frac{k^{2s+1}}{\mu ^{N+1}|\bar{x_i} -\mu ^{-1}y |^{N-2s}}\right) , \nonumber \\ {}&\quad l=1,..,N, \end{aligned}$$

(A.2)

and

$$\begin{aligned} \begin{aligned}&\partial _{N+1} U_{x_i,\Lambda }(y) - \partial _{N+1} PU_{x_i,\Lambda }(y) \\ {}&\quad = \frac{-(N-2s)A_{N,s} H(\bar{x_i},\mu ^{-1}y)}{2\Lambda ^{\frac{N-2s+2}{2}}\mu ^{N-2s}} + O\left( \frac{1}{\mu ^{N}|\bar{x_i} -\mu ^{-1} y |^{N} } +\frac{k^{2s}}{\mu ^{N}|\bar{x_i} -\mu ^{-1}y |^{N-2s}}\right) , \end{aligned}\nonumber \\ \end{aligned}$$

(A.3)

where $A_{N,s}$ is a constant only depend on N and s. If $y\in B_{{\frac{\mu r_3}{8k}}}(x_i)$, then

$$\begin{aligned} U_{x_i,\Lambda }(y) - PU_{x_i,\Lambda }(y)= & {} \frac{A_{N,s}H(\bar{x_i},\mu ^{-1}y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s}} + O\left( \frac{k^{N}}{\mu ^{N}}\right) , \end{aligned}$$

(A.4)

$$\begin{aligned} \partial _l U_{x_i,\Lambda }(y) - \partial _l PU_{x_i,\Lambda }(y)= & {} \frac{A_{N,s}\partial _l H(\bar{x_i},\mu ^{-1}y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s+1}} + O(\frac{k^{N+1}}{\mu ^{N+1}}), \quad l=1,..,N,\qquad \end{aligned}$$

(A.5)

and

$$\begin{aligned} \partial _{N+1} U_{x_i,\Lambda }(y) - \partial _{N+1} PU_{x_i,\Lambda }(y) = \frac{-(N-2s)A_{N,s} H(\bar{x_i},\mu ^{-1}y)}{2\Lambda ^{\frac{N-2s+2}{2}} \mu ^{N-2s}} + O\left( \frac{k^{N}}{\mu ^{N}}\right) .\qquad \end{aligned}$$

(A.6)

Proof

Let ${\bar{PU}}_{x_i,\Lambda }(x) = \mu ^{\frac{N-2s}{2}}PU_{x_i,\Lambda }(\mu x)$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}{\bar{PU}}_{x_i,\Lambda }(x) = \big (\mu ^{\frac{N-2s}{2}}U_{x_i,\Lambda }(\mu x)\big )^{2^{*}_{s}-1}, &{}\hbox { in } B_{1}(0), \\ \displaystyle {\bar{PU}}_{x_i,\Lambda }(x) = 0,&{} x \in B^{c}_{1}(0). \end{array}\right. } \end{aligned}$$

(A.7)

So,

$$\begin{aligned} \begin{aligned}&\mu ^{\frac{N-2s}{2}}U_{x_i,\Lambda }(\mu y) - {\bar{PU}}_{x_i,\Lambda }(y) \\&\quad = \int _{{\mathbb {R}}^{N}}\frac{C_{1,N,s}}{|x-y|^{N-2s}}\frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} }\\&\qquad - \int _{B_{1}(0)}G(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \\&\quad = \int _{{\mathbb {R}}^{N}\setminus B_{1}(0)}\frac{C_{1,N,s}}{|x-y|^{N-2s}}\frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \\ {}&\qquad + \int _{B_{1}(0)}H(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \\ {}&\quad =I_1 +I_2. \end{aligned} \end{aligned}$$

(A.8)

Case 1: $y\in B_{\frac{r_3}{8k}}(\bar{x_i})$, it is easy to check

$$\begin{aligned} |I_1| = O\left( \frac{k^{N}}{\mu ^{\frac{N+2s}{2} } }\right) , \end{aligned}$$

(A.9)

and

$$\begin{aligned} I_2 =&\int _{B_{1}(0)\setminus B_{\frac{r_0}{2k}}(\bar{x_i}) } H(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\ {}&+ \int _{ B_{\frac{r_0}{2k}(\bar{x_i})} }H(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\ =&\int _{ B_{\frac{r_0}{2k}(\bar{x_i})} }H(\bar{x_i},y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\&+ O\left( \int _{ B_{\frac{r_0}{2k}(\bar{x_i})} }|\bar{x_{i}} -x|^2|\nabla ^2 H(\bar{x_i} +t(\bar{x_i}-x),y) |\frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} }\right) \nonumber \\ {}&+O\left( \frac{k^{N}}{\mu ^{\frac{N+2s}{2} }}\right) \nonumber \\=&\frac{A_{N,s}H(\bar{x_i},y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{\frac{N-2s}{2}}} + O\left( \frac{k^{N}}{\mu ^{\frac{N+2s}{2} }}\right) . \end{aligned}$$

(A.10)

So,

$$\begin{aligned} \mu ^{\frac{N-2s}{2}}U_{x_i,\Lambda }(\mu y) - {\bar{PU}}_{x_i,\Lambda }(y) = \frac{A_{N,s}H(\bar{x_i},y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{\frac{N-2s}{2}}} + O(\frac{k^{N}}{\mu ^{\frac{N+2s}{2} }}). \end{aligned}$$

(A.11)

That is, if $y\in B_{{\frac{\mu r_3}{8k}}}({\bar{x}}_i)$, then

$$\begin{aligned} U_{x_i,\Lambda }(y) - PU_{x_i,\Lambda }(y) = \frac{A_{N,s}H(\bar{x_i},\mu ^{-1}y)}{\Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s}} + O\left( \frac{k^{N}}{\mu ^{N}}\right) . \end{aligned}$$

Case 2: $y \in B_{{\frac{r_3}{8k}}}(\bar{x_j})$, where $j\ne i$. In this case, it is easy to check

$$\begin{aligned} I_1 =O\left( \frac{1}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N} } +\frac{k^{2s}}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N-2s}}\right) , \end{aligned}$$

and

$$\begin{aligned} I_2&=\int _{B_{1}(0)\setminus B_{\frac{|y-\bar{x_i}|}{2}}(\bar{x_i}) } H(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\&\quad + \int _{ B_{\frac{|y-\bar{x_i}|}{2}(\bar{x_i}) } \cap B_{1}(0)}H(x,y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\&=\int _{ B_{\frac{r_3}{8k}(\bar{x_i})} }H(\bar{x_i},y) \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} } \nonumber \\&\quad + O( \int _{B_{\frac{|y-\bar{x_i}|}{2}(\bar{x_i})} }|\bar{x_{i}} -x|^2|\nabla ^2 H(\bar{x_i} +t(\bar{x_i}-x),y) |\frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} }) \nonumber \\&\quad +O\left( \int _{ B_{\frac{|y-\bar{x_i}|}{2}(\bar{x_i}) } \setminus B_{\frac{r_3}{8k}(\bar{x_i})} } |H(x,y)| \frac{C_{2,N,s}(\Lambda \mu )^{\frac{N+2s}{2}}}{ (1 + (\Lambda \mu |x-\bar{x_i}|)^{2})^{ \frac{N+2s}{2}} }\right) \nonumber \\&\quad +O\left( \frac{1}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N} }\right) \nonumber \\&=\frac{A_{N,s}H(\bar{x_i},y)}{\Lambda ^{\frac{N-2s}{2} }\mu ^{\frac{N-2s}{2}}} + O(\frac{1}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N} }+\frac{k^{2s}}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N-2s}}). \end{aligned}$$

(A.12)

So,

$$\begin{aligned} \begin{aligned}&\mu ^{\frac{N-2s}{2}}U_{x_i,\Lambda }(\mu y) - {\bar{PU}}_{x_i,\Lambda }(y) \\ {}&\quad = \frac{A_{N,s}H(\bar{x_i},y)}{\Lambda ^{\frac{N-2s}{2} }\mu ^{\frac{N-2s}{2}}} + O \left( \frac{1}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N} }+\frac{k^{2s}}{\mu ^{\frac{N+2s}{2}}|\bar{x_i} -y |^{N-2s}}\right) . \end{aligned} \end{aligned}$$

(A.13)

That is, if $y\in B_{{\frac{\mu \min (r_0,1)}{8k}}}(x_j)$, then

$$\begin{aligned}&U_{x_i,\Lambda }(y) - PU_{x_i,\Lambda }(y) \nonumber \\&\quad = \frac{A_{N,s}H(\bar{x_i},\mu ^{-1}y)}{ \Lambda ^{\frac{N-2s}{2}}\mu ^{N-2s}} + O \left( \frac{1}{\mu ^{N}|\bar{x_i} -\mu ^{-1} y |^{N} } +\frac{k^{2s}}{\mu ^{N}|\bar{x_i} -\mu ^{-1}y |^{N-2s}}\right) . \end{aligned}$$

(A.14)

Since for $l=1,...,N+1$,

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}\partial _l PU_{x_i,\Lambda }(x) = \partial _l (U_{x_i,\Lambda }(\mu x))^{2^{*}_{s}-1}, &{}\hbox { in } B_{1}(0), \\ \displaystyle {\bar{PU}}_{x_i,\Lambda }(x) = 0, &{}x \in B^{c}_{1}(0). \end{array}\right. } \end{aligned}$$

(A.15)

Similar to (A.1) and (A.2), we can prove (A.3)-(A.6). $\square$

Appendix B. Energy expansion

In this section, we will give the expansion of the energy $I(W_{r,\Lambda })$. Recall that $\mu =k^{\frac{N-2s+1}{N-2s}}$, $d=1-\frac{r}{\mu }$,

$$\begin{aligned} I(u)&=\frac{1}{2}\int _{B_{\mu }(0)}|(-\Delta )^{\frac{s}{2}} u|^{2} -\frac{1}{2^{*}_s}\int _{B_{\mu }(0)}K\left( \frac{|y|}{\mu }\right) |u|^{2^{*}_s},\\ U_{x_j,\Lambda }(y)&= C_{N,s}\frac{\Lambda ^{\frac{N-2s}{2}}}{(1+\Lambda ^{2}|y-x_j|^{2})^{\frac{N-2s}{2}}}, \end{aligned}$$

and

$$\begin{aligned} W_{r,\Lambda }(y)=\sum _{j=1}^{k}PU_{x_j,\Lambda }(y), \end{aligned}$$

where $PU_{x_j,\Lambda }(y)$ is the solution of (1.4).

Proposition B.1

If $N\ge 3$ for $\frac{3}{4} \le s<1$ and $3\le N < 2s -1 +\frac{2}{3-4s}$ for $\frac{11-\sqrt{41}}{8}< s<\frac{3}{4}$, then we have

$$\begin{aligned} I(W_{r,\Lambda })= & {} k\big (A+\frac{B_{1}H(\bar{x_1},\bar{x_1})}{\Lambda ^{N-2s}\mu ^{N-2s}} +B_2K'(1)d -\sum _{j=2}^{k}\frac{B_{1}G(\bar{x_j},\bar{x_1})}{\Lambda ^{N-2s}\mu ^{N-2s}} + O\big (\frac{1}{\mu ^{1+\sigma }}\big ) \big )\nonumber \\= & {} k\big ( A + \frac{A_1}{\Lambda ^{N-2s}\mu ^{N-2s}d^{N-2s}} +A_2 d -\frac{A_3k^{N-2s}}{\Lambda ^{N-2s}\mu ^{N-2s}} \nonumber \\&\times \sum _{i=2}^{+\infty }\frac{1}{((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} \!+\!1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv \!+\! O\big (\frac{1}{\mu ^{1+\sigma }}\big )\big ),\qquad \nonumber \\ \end{aligned}$$

(B.1)

where A, $A_1$, $A_2$, $A_3$, $B_1$ and $B_2$ are positive constants.

Proof

By the symmetry, we have

$$\begin{aligned} \begin{aligned}&\int _{B_{\mu }(0)}|(-\Delta )^{\frac{s}{2}} W_{r,\Lambda ]}|^{2}\\&\quad =\sum _{j=1}^{k}\sum _{i=1}^{k}\int _{B_{\mu }(0)}U_{x_j,\Lambda }^{2^{*}_s-1}PU_{x_i,\Lambda } \\ {}&\quad = k\big ( \int _{B_{\mu }(0)} U_{x_1,\Lambda }^{2^{*}_s} -\int _{B_{\mu }(0)} U_{x_1,\Lambda }^{2^{*}_s-1}(U_{x_1,\Lambda }-PU_{x_1,\Lambda } ) +\sum _{i=2}^{k}\int _{B_{\mu }(0)} U_{x_1,\Lambda }^{2^{*}_s-1}PU_{x_i,\Lambda } \big ) \\ {}&\quad = k\big ( \int _{{\mathbb {R}}^{N}} U_{x_1,\Lambda }^{2^{*}_s} - \frac{\bar{B_1}H(\bar{x_1}, \bar{x_1} )}{(\Lambda \mu )^{N-2s}} + \frac{\bar{B_1}\sum _{i=2}^{k}G( \bar{x_i}, \bar{x_1} )}{(\Lambda \mu )^{N-2s}} +O\left( \frac{1}{\mu ^{1+\sigma }}\right) \big ), \end{aligned}\nonumber \\ \end{aligned}$$

(B.2)

where $\bar{B_1} =\displaystyle \int _{{\mathbb {R}}^{N}} U_{x_1,\Lambda }^{2^{*}_s-1}.$

Let

$$\begin{aligned} \Omega _j =\left\{ y:\hbox { }y=(y',y'')\in B_{\mu }(0),\hbox { } \left\langle \frac{y'}{|y'|},\frac{x_j}{|x_j|} \right\rangle \ge \cos \frac{\pi }{k} \right\} . \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned}&\int _{B_{\mu }(0)}K\big (\frac{|y|}{\mu }\big ) |W_{r,\Lambda }|^{2^{*}_s}\\&\quad =k\int _{\Omega _1} K\big (\frac{|y|}{\mu }\big )|W_{r,\Lambda }|^{2^{*}_s} \\ {}&\quad =k\big ( \int _{\Omega _1}K\big (\frac{|y|}{\mu }\big ) (PU_{x_1,\Lambda })^{2^{*}_s} -2^{*}_s \int _{\Omega _1}\sum _{i=2}^{k}(PU_{x_{1,\Lambda }})^{2^{*}_s-1}PU_{x_i,\Lambda } \\ {}&\qquad +O\big (\int _{\Omega _1}\big | K\big (\frac{|y|}{\mu }\big )-1 \big | \sum _{i=2}^{k}U_{x_{1,\Lambda }}^{2^{*}_s-1}U_{x_i,\Lambda } \\ {}&\qquad + \int _{\Omega _1} (U_{x_{1,\Lambda }})^{2^{*}_s-2} \big (\sum _{i=2}^{k}U_{x_i,\Lambda }\big )^{2} + \int _{\Omega _1} \big (\sum _{i=2}^{k}U_{x_i,\Lambda }^{2^{*}_s} \big ) \big ). \end{aligned} \end{aligned}$$

(B.3)

Note that for $y \in \Omega _1$, $|y-x_i|\ge |y-x_1|$. Thus

$$\begin{aligned} \begin{aligned} (U_{x_{1,\Lambda }})^{2^{*}_s-2}\left( \sum _{i=2}^{k}U_{x_i,\Lambda }\right) ^{2} \le \frac{C}{(1+|y-x_1|)^{N+\theta }}\sum _{i=2}^{k}\frac{1}{|x_i-x_1|^{N-\theta }}, \end{aligned} \end{aligned}$$

(B.4)

and

$$\begin{aligned} \begin{aligned} \left( \sum _{i=2}^{k}U_{x_i,\Lambda }\right) ^{2^{*}_s} \le \frac{C}{(1+|y-x_1|)^{N+\theta }}\sum _{i=2}^{k}\frac{1}{|x_i-x_1|^{N-\theta }}. \end{aligned} \end{aligned}$$

(B.5)

where $\theta$ is a small positive constant. We can chose $\theta$ small enough, then

$$\begin{aligned} \int _{\Omega _1} (U_{x_{1,\Lambda }})^{2^{*}_s-2}\big (\sum _{i=2}^{k}U_{x_i,\Lambda }\big )^{2} + \int _{\Omega _1} \big (\sum _{i=2}^{k}U_{x_i,\Lambda }\big )^{2^{*}_s} = O\big (\big (\frac{k}{\mu }\big )^{N-\theta }\big ) =O\big ( \frac{1}{\mu ^{1+\sigma }}\big ).\qquad \end{aligned}$$

(B.6)

On the other hand, it is easy to show that

$$\begin{aligned} \begin{aligned} \int _{\Omega _1}\sum _{i=2}^{k}(PU_{x_{1,\Lambda }})^{2^{*}_s-1}PU_{x_i,\Lambda }&= \sum _{i=2}^{k}\frac{{\bar{B}}_1G(\bar{x_i},\bar{x_1} )}{(\Lambda \mu )^{N-2s}} +O\big (\big (\frac{k}{\mu }\big )^{N}\big ) \\ {}&=\sum _{i=2}^{k}\frac{{\bar{B}}_1G(\bar{x_i},\bar{x_1} )}{(\Lambda \mu )^{N-2s}}+O\big ( \frac{1}{\mu ^{1+\sigma }}\big ), \end{aligned} \end{aligned}$$

(B.7)

and

$$\begin{aligned} \begin{aligned} \int _{\Omega _1}| K\big (\frac{|y|}{\mu }\big )-1 | \sum _{i=2}^{k}U_{x_{1,\Lambda }}^{2^{*}_s-1}U_{x_i,\Lambda } = O\big ( \frac{1}{\mu ^{1+\sigma }}\big ). \end{aligned} \end{aligned}$$

(B.8)

Moreover

$$\begin{aligned} \begin{aligned}&\int _{\Omega _1}K\big (\frac{|y|}{\mu }\big )\big (PU_{x_1,\Lambda }\big )^{2^{*}_s} \\&\quad = \int _{B_{\frac{r_3 \mu }{8k}}(x_2)}K\big (\frac{|y|}{\mu }\big )\big (PU_{x_1,\Lambda }\big )^{2^{*}_s} + O\big ( \frac{1}{\mu ^{1+\sigma }}\big ) \\&\quad =\int _{B_{\frac{r_3 \mu }{8k}}(x_2)}(PU_{x_1,\Lambda })^{ 2^{*}_s }+ \int _{B_{\frac{r_3 \mu }{8k}}(x_2)}\big (K\big (\frac{|y|}{\mu }\big ) -1 \big )U_{x_1,\Lambda }^{ 2^{*}_s } \\ {}&\qquad +O\big ( \int _{B_{\frac{r_3 \mu }{8k}}(x_2)} \big |K\big (\frac{|y|}{\mu }\big ) -1\big |U_{x_1,\Lambda }^{ 2^{*}_s -1}\frac{H({\bar{x}}_{1},\mu ^{-1}y)}{\mu ^{N-2s}} \big ) + O\big ( \frac{1}{\mu ^{1+\sigma }}\big ) \\ {}&\quad =\int _{{\mathbb {R}}^{N}}U_{0,1}^{ 2^{*}_s } -2^{*}_s\frac{{\bar{B}}_1H({\bar{x}}_1,{\bar{x}}_1)}{ (\Lambda \mu )^{N-2s} } + \int _{B_{\frac{r_3 \mu }{8k}}(x_2)}\big (K\big (\frac{|y|}{\mu }\big ) -1 \big )U_{x_1,\Lambda }^{ 2^{*}_s } +O\big ( \frac{1}{\mu ^{1+\sigma }}\big ). \end{aligned} \end{aligned}$$

(B.9)

However,

$$\begin{aligned} \begin{aligned} \int _{B_{\frac{r_3 \mu }{8k}}(x_2)}\big (K\big (\frac{|y|}{\mu }\big ) -1 \big )U_{x_1,\Lambda }^{ 2^{*}_s }&= \big ( K(|{\bar{x}}_1|) -1 \big )\int _{{\mathbb {R}}^{N}}U_{0,1}^{ 2^{*}_s } + O\big (\frac{1}{k^{2}}\big ) \\ {}&= -K'(1)d\int _{{\mathbb {R}}^{N}}U_{0,1}^{ 2^{*}_s } + O\big ( \frac{1}{\mu ^{1+\sigma }}\big ). \end{aligned} \end{aligned}$$

(B.10)

Thus, we have proved

$$\begin{aligned} \begin{aligned} \int _{B_{\mu }(0)} K\big (\frac{|y|}{\mu }\big ) |W_{r,\Lambda }|^{2^{*}_s}&= k\big (\int _{{\mathbb {R}}^{N}}U_{0,1}^{ 2^{*}_s } -K'(1)d\int _{{\mathbb {R}}^{N}}U_{0,1}^{ 2^{*}_s } - 2^{*}_s \frac{{\bar{B}}_1H({\bar{x}}_1,{\bar{x}}_1)}{ (\Lambda \mu )^{N-2s} } \\&\quad + \sum _{i=2}^{k}\frac{{\bar{B}}_1G({\bar{x}}_i,{\bar{x}}_1)}{ (\Lambda \mu )^{N-2s} } + O\big ( \frac{1}{\mu ^{1+\sigma }}\big ) \big ). \end{aligned} \end{aligned}$$

(B.11)

Combining (B.2) and (B.11), we can get

$$\begin{aligned} \begin{aligned} I(W_{r,\Lambda })=k\big ( A+\frac{B_{1}H(\bar{x_1},\bar{x_1})}{\Lambda ^{N-2s}\mu ^{N-2s}} +B_2K'(1)d -\sum _{i=2}^{k}\frac{B_{1}G(\bar{x_i},\bar{x_1})}{\Lambda ^{N-2s}\mu ^{N-2s}} + O(\frac{1}{\mu ^{1+\sigma }})\big ), \end{aligned}\nonumber \\ \end{aligned}$$

(B.12)

where A, $B_1$ and $B_2$ are positive constants.

Now, we estimate $H( {\bar{x}}_1,{\bar{x}}_1)$ and $G( {\bar{x}}_i,{\bar{x}}_1)$, $i\ge 2$. Let ${\bar{x}}_1^{*} = (\frac{1}{1-d},0,...,0 )$ be the reflection of ${\bar{x}}_1$ withe respect to the unit sphere. Then

$$\begin{aligned} H( {\bar{x}}_1,{\bar{x}}_1) = \frac{1}{2^{N-2s}d^{N-2s}}\big (1+O(d)\big ). \end{aligned}$$

(B.13)

We can compute that,

$$\begin{aligned}&\sum _{i=2}^{k} G( {\bar{x}}_i,{\bar{x}}_1) \nonumber \\&\quad = C\sum _{i=2}^{[\frac{k}{2}]}\frac{1}{\big (2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{N-2s}} \int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( 2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{2} } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv\nonumber \\&\qquad +O(k^{N-2s-1})\nonumber \\&\quad = Ck^{N-2s}\sum _{i=2}^{+\infty }\frac{1}{2^{N-2s}((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv \nonumber \\&\qquad + O(k^{N-2s-1}). \end{aligned}$$

(B.14)

In fact, for $i\le k^{\alpha }$, where $\alpha \in (\frac{1}{N-2s-1},1)$ is a fix constant, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{\big (2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{N-2s}}\\&\quad = \frac{k^{N-2s}}{ (1-d)^{N-2s}2^{N-2s}((i-1)\pi )^{N-2s}}+ {\bar{O}}\big (\frac{k^{N-2s-2}}{(i-1)^{N-2s-2}}\big ) \\ {}&\quad =\frac{k^{N-2s}}{ 2^{N-2s}((i-1)\pi )^{N-2s}}+ {\bar{O}}\big (\frac{k^{N-2s-2}}{(i-1)^{N-2s-2}}\big ) +{\bar{O}}\big (\frac{k^{N-2s-1}}{(i-1)^{N-2s}}\big ) , \end{aligned} \end{aligned}$$

(B.15)

where ${\bar{O}}(f(i,k))$ means that, there is a constant C and $k_0$ , for any $k >k_0$ and any $2 \le i \le k^{\alpha }$,

$$\begin{aligned} | {\bar{O}}(f(i,k)) | \le C |f(i,k)|. \end{aligned}$$

On the other hand,

$$\begin{aligned}&\int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( 2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{2} } + \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv \\ {}&\quad =\int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( 2( 1-d ) \big (\frac{(i-1)\pi }{k} + {\bar{O}}\big (\frac{(i-1)^3)}{k^3}\big ) \big ) \big )^{2} } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv \\ {}&\quad =\int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( \big (2( 1-d ) \big (\frac{(i-1)\pi }{k}\big )^2 + {\bar{O}} \big (\frac{(i-1)^4)}{k^4}\big ) \big ) \big ) } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv \\ {}&\quad =\int _{1}^{ \big ( \frac{ d^{2} }{ \big (\frac{(i-1)\pi }{k})^2 \big ) } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv +{\bar{O}}\big (\frac{(i-1)^{2(1-s)}}{k^2}\big ) +{\bar{O}}\big (\frac{1}{(i-1)^{2s}k} \big ). \end{aligned}$$

Noting $N \ge 3$, by direct computation, we have

$$\begin{aligned} \begin{aligned}&\sum _{i=2}^{[k^{\alpha }]}\frac{1}{\big (2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{N-2s}} \int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( 2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{2} } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv \\ {}&\quad =k^{N-2s}\sum _{i=2}^{+\infty }\frac{1}{2^{N-2s}((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv + O(k^{N-2s-1}) \\ {}&\qquad +O\big (\sum _{i=2}^{[k^{\alpha }]}\frac{1}{(i-1)^{N-2s}}\big ) \\ {}&\quad =k^{N-2s}\sum _{i=2}^{+\infty }\frac{1}{2^{N-2s}((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv + O(k^{N-2s-1}) \\ {}&\qquad +O(k^{(N-2s)-\alpha (N-2s-1)}) \\ {}&\quad =k^{N-2s}\sum _{i=2}^{+\infty }\frac{1}{2^{N-2s}((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv + O(k^{N-2s-1}). \end{aligned} \end{aligned}$$

(B.16)

Since

$$\begin{aligned} \sin \big (\frac{i-1}{k}\big ) \ge C|\frac{i-1}{k}|,\hbox { for } i=1,...,\big [\frac{k}{2}\big ], \end{aligned}$$

and

$$\begin{aligned} \int _1^{+\infty }(v^2-1)^{s-1}v^{1-N} dv < +\infty , \end{aligned}$$

it is easy to check

$$\begin{aligned}&\sum _{[k^{\alpha }]+1}^{\big [\frac{k}{2}\big ]}\frac{1}{\big (2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{N-2s}} \int _{1}^{ \big ( \frac{ d^{2}(2-d)^2 }{ \big ( 2( 1-d )\sin \frac{(i-1)\pi }{k} \big )^{2} } +1 \big )^{\frac{1}{2}} } (v^2-1)^{s-1}v^{1-N} dv\nonumber \\&\quad = O(k^{(N-2s)-\alpha (N-2s-1)}) =O(k^{N-2s-1}). \end{aligned}$$

(B.17)

Combining (B.16) and (B.17), we have (B.14).

Finally, combining (B.12), (B.13) and (B.14), we get (B.1).$\square$

Proposition B.2

If $N\ge 3$ for $\frac{3}{4} \le s<1$ and $3\le N < 2s -1 +\frac{2}{3-4s}$ for $\frac{11-\sqrt{41}}{8}< s<\frac{3}{4}$, then we have

$$\begin{aligned} \begin{aligned} \frac{\partial I(W_{r,\Lambda })}{\partial \Lambda }&=kB_1(N-2s)\big ( -\frac{ H(\bar{x_1},\bar{x_1})}{\Lambda ^{N-2s+1}\mu ^{N-2s}} +\sum _{j=2}^{k}\frac{ G(\bar{x_j},\bar{x_1})}{\Lambda ^{N-2s+1}\mu ^{N-2s}} + O\big (\frac{1}{\mu ^{1+\sigma }}\big )\big ) \\&= k \big ( -\frac{A_1(N-2s)}{\Lambda ^{N-2s+1}\mu ^{N-2s}d^{N-2s}} + \frac{A_3k^{N-2s}(N-2s)}{\Lambda ^{N-2s+1}\mu ^{N-2s}} \\&\quad \times \sum _{i=2}^{+\infty }\frac{1}{((i-1)\pi )^{N-2s}}\int _{1}^{\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{1}{2}}} (v^2-1)^{s-1}v^{1-N} dv + O\big (\frac{1}{\mu ^{1+\sigma }}\big )\big ) , \end{aligned} \end{aligned}$$

(B.18)

and

$$\begin{aligned} \begin{aligned} \frac{\partial I(W_{r,\Lambda })}{\partial d}&=k\big ( \frac{B_{1}\frac{\partial H(\bar{x_1},\bar{x_1})}{ \partial d}}{\Lambda ^{N-2s}\mu ^{N-2s}} +B_2K'(1)-\sum _{j=2}^{k}\frac{B_1\frac{\partial G(\bar{x_j},\bar{x_1})}{ \partial d}}{{\Lambda ^{N-2s}\mu ^{N-2s}} }+ O(\frac{1}{\mu ^{\sigma }})\big ) \\&= k \big ( -\frac{A_1(N-2s)}{\Lambda ^{N-2s}\mu ^{N-2s}d^{N-2s+1}} +A_2 - \frac{A_3k^{N-2s}(N-2s)}{\Lambda ^{N-2s}\mu ^{N-2s}} \\&\quad \times \sum _{i=2}^{+\infty }\frac{1}{((i-1)\pi )^{N}}\big ( \frac{(dk)^{2}}{((i-1)\pi )^{2}} +1 \big )^{\frac{-N}{2}}dk^{2}(dk)^{2(s-1)} + O\big (\frac{1}{\mu ^{\sigma }} \big )\big ). \end{aligned} \end{aligned}$$

(B.19)

where $A_1$, $A_2$, $A_3$, $B_1$ and $B_2$ are the same positive constants as in Proposition B.1.

Proof

We use $\partial$ to denote either $\frac{\partial }{\partial \Lambda }$ or $\frac{\partial }{\partial d}$. Using the symmetry, we have

$$\begin{aligned} \partial I(W_{r,\Lambda })&= k\big (\frac{2^{*}_s-1}{2}\int _{B_{\mu }(0)}\sum _{i=1}^{k}U_{x_1,\Lambda }^{ 2^{*}_s-2 }\partial U_{x_1,\Lambda } PU_{x_i,\Lambda } + \frac{1}{2}\int _{B_{\mu }(0)}\sum _{i=1}^{k}U_{x_1,\Lambda }^{ 2^{*}_s-1 }\partial PU_{x_i,\Lambda } \\&\quad - \int _{\Omega _1}K\big (\frac{|y|}{\mu }\big ) W_{r,\Lambda }^{ 2^{*}_s-1} \partial W_{r,\Lambda } \big ). \end{aligned}$$

Then the proof of this proposition is similar to the proof of Proposition B.1, so we omit it. $\square$

Appendix C Basic estimates

In this section, we list some lemmas, whose proof can be found in [9] and [22].

For each fixed i and j, $i \ne j$, consider the function

$$\begin{aligned} g_{i,j}(y) = \frac{1}{(1+|y-x_i|)^{\alpha }}\frac{1}{(1+|y-x_j|)^{\beta }}, \end{aligned}$$

(C.1)

where $\alpha \ge 1$ and $\beta \ge 1$ are constants.

Lemma C.1

For any constant $0< \sigma <\min (\alpha ,\beta )$, there is a constant $C>0$, such that

$$\begin{aligned} g_{i,j}(y)\le \frac{C}{|x_i-x_j|^{\sigma }}\left( \frac{1}{(1+|y-x_i|)^{\alpha +\beta -\sigma }} +\frac{1}{(1+|y-x_j|)^{\alpha +\beta -\sigma }} \right) . \end{aligned}$$

Lemma C.2

For any constant $0< \sigma <N-2s$, there is a constant $C>0$, such that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\frac{1}{|y-z|^{N-2s}}\frac{1}{(1+|z|)^{2s+\sigma }}dz \le \frac{C}{(1+|y|)^{\sigma }}. \end{aligned}$$

Lemma C.3

Suppose that $N\ge 3$, then there is a small $\theta >0$, such that

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}\frac{1}{|y-z|^{N-2s}}W^{2^*_s-2}_{r,\Lambda }(z)\sum _{j=1}^k \frac{1}{(1+|z-x_j|)^{\frac{N-2s}{2}+\tau }}dz \\&\quad \le C\sum _{j=1}^k\frac{1}{(1+|y-x_j|)^{\frac{N+2s}{2}+\tau +\theta }}, \end{aligned}$$

where $W_{r,\Lambda } = \sum _{j=1}^k PU_{x_j,\Lambda }.$