Non-radial solutions for higher order Hénon-type equation with critical exponent

Abstract

We consider the following polyharmonic equation with critical exponent

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{m} u = K(|y|)u^{m^*-1},u>0&{}\hbox {in } B_{1}(0), \\ \displaystyle u \in \mathscr {D}_0^{m,2}(B_1(0)), \end{array}\right. } \end{aligned}$$

where $m>0$ is a integer, $m^*=:\frac{2N}{N-2m}$, $B_{1}(0)$ is the unit ball in $\mathbb {R}^{N}$, $N \ge 2m+4$, $K:[0,1] \rightarrow \mathbb {R}^{N}$ is a bounded function, $K'(1)>0$ and $K''(1)$ exists. We prove a non-degeneracy result of the non-radial solutions constructed in Guo and Li (Calc Var PDEs 46(3–4):809–836, 2013) via the local Pohozaev identities for $N \ge 2m+4$. Then we apply the non-degeneracy result to obtain new existence of non-radial solutions for $N \ge 6m$.

Appendices

Appendix A. The estimate of U-PU

In the following, we always assume that

$$\begin{aligned} x_j=\bigg ( r \cos \frac{2(j-1)\pi }{k},r \sin \frac{2(j-1)\pi }{k}, \textbf{0} \bigg ),\quad j=1,\ldots k, \end{aligned}$$

$r\in [ \mu (1-\frac{r_0}{k}),\mu (1- \frac{r_1}{k})]$. Let ${\bar{x}}_j=\frac{1}{\mu }x_j$, G(x, y) be the Green function of $(-\Delta )^m$ in $B_1(0)$ with homogenous Dirichlet boundary condition, and H(x, y) be the regular part of Green function. We use $PU_{x_j,\Lambda _k \mu _k}$ to denote the solution of (1.6) and $r_3=min (r_0,1)$.

For $l=1,\ldots , N$, denote

$$\begin{aligned} \partial _l PU_{x,\Lambda \mu }=\frac{ \partial PU_{x,\Lambda \mu }(y)}{\partial x_l},\quad \partial _l U_{x,\Lambda \mu }=\frac{ \partial U_{x,\Lambda \mu }(y)}{\partial x_l}; \end{aligned}$$

for $l=N+1$, we set

$$\begin{aligned} \partial _l PU_{x,\Lambda \mu }=\frac{ \partial PU_{x,\Lambda \mu }(y)}{\partial \mu },\quad \partial _l U_{x,\Lambda \mu }=\frac{ \partial U_{x,\Lambda \mu }(y)}{\partial \mu }. \end{aligned}$$

Lemma A.1

Assume $N\ge 2m+4$, for any $i=1,\ldots ,k$, if $y\in B_{\frac{\mu r_3}{8k}(x_j)}$ and $j\ne i$, then

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

(A.1)

$$\begin{aligned}{} & {} \partial _l U_{x_i,\Lambda \mu }(y)-\partial _l PU_{x_i,\Lambda \mu }(y)\nonumber \\{} & {} \quad =\frac{A_{N,m}\partial _l H( x_i,y)}{\Lambda ^{\frac{N-2m}{2}}\mu ^{\frac{N-2m}{2}}} +O\left( \frac{1}{\mu ^{\frac{N-2m}{2}+2}|x_i -y |^{N-2m+3} }+\frac{k^{2m+1}}{\mu ^{\frac{N+2m}{2}}|x_i -y |^{N-2m}}\right) ,\nonumber \\{} & {} \qquad \hbox { for } l=1,\ldots ,N, \end{aligned}$$

(A.2)

and

$$\begin{aligned}{} & {} \partial _{N+1} U_{x_i,\Lambda \mu }(y)-\partial _{N+1} PU_{x_i,\Lambda \mu }(y)\nonumber \\{} & {} \quad =\frac{-(N-2m)A_{N,m}H( x_i,y)}{2\Lambda ^{\frac{N-2m}{2}}\mu ^{\frac{N-2m}{2}+1} }+O\left( \frac{1}{\mu ^{\frac{N-2m}{2}+3}|x_i -y |^{N-2m+2} }\right. \nonumber \\{} & {} \qquad \left. +\frac{k^{2m}}{\mu ^{\frac{N+2m}{2}+1}|x_i -y |^{N-2m}}\right) , \end{aligned}$$

(A.3)

where $A_{N,m}$ is a constant only depend on N and m.

If $y\in B_{\frac{\mu r_3}{8k}}(x_i)$, then

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

(A.4)

$$\begin{aligned}{} & {} \partial _l U_{x_i,\Lambda \mu }(y)-\partial _lPU_{x_i,\Lambda \mu }(y)=\frac{A_{N,m}\partial _l H(x_i,y)}{\Lambda ^{\frac{N-2m}{2}}\mu ^{\frac{N-2m}{2}}}\nonumber \\{} & {} \quad +O\left( \frac{k^{N-2m+3}}{\mu ^{\frac{N-2m}{2}+2}}\right) , \text { for } l=1,\ldots , N, \end{aligned}$$

(A.5)

and

$$\begin{aligned}{} & {} \partial _{N+1} U_{x_i,\Lambda \mu }(y)-\partial _{N+1} PU_{x_i,\Lambda \mu }(y)=\frac{-(N-2m)A_{N,m}H( x_i,y)}{2\Lambda ^{\frac{N-2m}{2}}\mu ^{\frac{N-2m}{2}+1} }\nonumber \\{} & {} \quad +O\left( \frac{k^{N-2m+2}}{\mu ^{\frac{N-2m}{2}+3}}\right) . \end{aligned}$$

(A.6)

Proof

By the potential theory,

$$\begin{aligned} \begin{aligned}&U_{x_i,\Lambda \mu } - PU_{x_i,\Lambda \mu }(y) \\&\quad = \int _{\mathbb {R}^{N}}\frac{C_{1,N,m}}{|x-y|^{N-2m}}\frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} }\\&\qquad - \int _{B_{1}(0)}G(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \\ {}&\quad = \int _{\mathbb {R}^{N}\setminus B_{1}(0)}\frac{C_{1,N,m}}{|x-y|^{N-2m}}\frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \\ {}&\qquad + \int _{B_{1}(0)}H(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \\ {}&\quad =M_1 +M_2. \end{aligned} \end{aligned}$$

(A.7)

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

$$\begin{aligned} |M_1| = O\left( \frac{k^{N}}{\mu ^{\frac{N+2m}{2} } }\right) , \end{aligned}$$

(A.8)

and

$$\begin{aligned} M_2= & {} \int _{B_{1}(0)\setminus B_{\frac{r_0}{2k}}(x_i) } H(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\{} & {} + \int _{ B_{\frac{r_0}{2k}(x_i)} }H(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\= & {} \int _{ B_{\frac{r_0}{2k}(x_i)} }H(x_i,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\{} & {} + O\left( \int _{ B_{\frac{r_0}{2k}(x_i)} }|x_{i} -x|^2|\nabla ^2 H(x_i +t(x_i-x),y) |\frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} }\right) \nonumber \\{} & {} +O\left( \frac{k^{N}}{\mu ^{\frac{N+2m}{2} }}\right) =\frac{A_{N,m}H(x_i,y)}{\Lambda ^{\frac{N-2m}{2}}\mu ^{\frac{N-2m}{2}}} + O\left( \frac{k^{N-2m+2}}{\mu ^{\frac{N-2m}{2}+2 }}\right) . \end{aligned}$$

(A.9)

So,

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

(A.10)

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

$$\begin{aligned} M_1 =O\left( \frac{1}{\mu ^{\frac{N+2m}{2}}|x_i -y |^{N} } +\frac{k^{2m}}{\mu ^{\frac{N+2m}{2}}|x_i -y |^{N-2m}}\right) , \end{aligned}$$

and

$$\begin{aligned} M_2= & {} \int _{B_{1}(0)\setminus B_{\frac{|y-x_i|}{2}}(x_i) } H(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\{} & {} + \int _{ B_{\frac{|y-x_i|}{2}(x_i) } \cap B_{1}(0)}H(x,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\= & {} \int _{ B_{\frac{r_3}{8k}(x_i)} }H(x_i,y) \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} } \nonumber \\{} & {} + O\left( \int _{B_{\frac{|y-x_i|}{2}(x_i)} }|x_i-x|^2|\nabla ^2 H(x_i +t(x_i-x),y) |\frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} }\right) \nonumber \\{} & {} +O\left( \int _{ B_{\frac{|y-x_i|}{2}(x_i) } \setminus B_{\frac{r_3}{8k}(x_i)} } |H(x,y)| \frac{C_{2,N,m}(\Lambda \mu )^{\frac{N+2m}{2}}}{ (1 + (\Lambda \mu |x-x_i|)^{2})^{ \frac{N+2m}{2}} }\right) \nonumber \\{} & {} +O\left( \frac{1}{\mu ^{\frac{N+2m}{2}}|x_i -y |^{N} }\right) \nonumber \\= & {} \frac{A_{N,m}H(x_i,y)}{\Lambda ^{\frac{N-2m}{2} }\mu ^{\frac{N-2m}{2}}} + O\left( \frac{1}{\mu ^{\frac{N-2m}{2}+2}|x_i -y |^{N-2m+2} }+\frac{k^{2m}}{\mu ^{\frac{N+2m}{2}}|x_i -y |^{N-2m}}\right) .\nonumber \\ \end{aligned}$$

(A.11)

So

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

(A.12)

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

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

(A.13)

Similar to (A.1 ), (A.2), we can prove (A.4)–(A.6). $\square$

For the completeness of the proof, we supplement the estimation of the derivative of H. In area $B_1(0)$ the Green function for the Dirichlet problem is positive and given by

$$\begin{aligned} G(x,y)=k_{m,N}\frac{1}{|x-y|^{N-2m}}\int _{1}^{\frac{ [xy]}{|x-y|}}\frac{(v^2-1)^{m-1}}{v^{N-1}}dv, \end{aligned}$$

where

$$\begin{aligned}{}[xy]=\big ||x|y-\frac{x}{|x|}\big |,k_{m,N}=\frac{1}{w_n4^{m-1}}((m-1)!)^2, \end{aligned}$$

$w_n$ is the n-dimensional unit sphere surface area. Using the following identity

$$\begin{aligned} \int _{1}^\infty \frac{(v^2-1)^{m-1}}{v^{N-1}}dv=\frac{2^{m-1}(m-1)!}{(N-2)\ldots (N-2m)}, \end{aligned}$$

and (B.13) we can get the expression of H

$$\begin{aligned} H(x,y) = \Gamma (x,y)-G(x,y)=k_{m,N}\frac{1}{|x-y|^{N-2m}}\int ^{\infty }_{\frac{ [xy]}{|x-y|}}\frac{(v^2-1)^{m-1}}{v^{N-1}}dv, \end{aligned}$$

(A.14)

we have

Lemma A.2

The above function H is satisfied

$$\begin{aligned} |\frac{\partial ^{\alpha } H(x,y)}{\partial ^{\alpha _1} x_1\ldots \partial ^{\alpha _N} x_N}|\le \frac{C_{m,N}}{[xy]^{N-2m+\alpha }}, \quad \alpha \ge 0 \quad \alpha _1+\cdots +\alpha _s=\alpha . \end{aligned}$$

(A.15)

Proof

By definition after a variable substitution, we have

$$\begin{aligned} \begin{aligned} H(x,y)=&\frac{k_{m,N}}{[xy]^{N-2m}}\int _{1}^\infty \frac{(v^2-\frac{|x-y|^2}{[xy]^2})^{m-1}}{v^{N-1}}dv\\ \le&\frac{k_{m,N}}{[xy]^{N-2m}} \int _{1}^\infty \frac{1}{v^{N-2m+1}}dv\\ \le&C\frac{1}{[xy]^{N-2m}}, \end{aligned} \end{aligned}$$

repeated derive the function and similarity calculation lead us to the result.

$\square$

Appendix B. Pohozaev indentities and the estimates of Green’s function

In this part, we will establish the Pohozaev indentities for polyharmonic operator in the small domain $B_{\frac{\delta }{k_n}}(x_{k_n,1})$, where $\delta >0$ is a small fixed constant. And then we give some expression related to Green’s function. We will use the same notations as before.

Recall that the equation and its linearized equation read as:

$$\begin{aligned} (-\Delta )^m u = K(|y|) u^{m^*-1},\text { in } B_1(0); \end{aligned}$$

(B.1)

and

$$\begin{aligned} (-\Delta )^m\xi =(m^*-1)K(|y|)u^{m^*-2}\xi , \text { in } B_1(0). \end{aligned}$$

(B.2)

Lemma B.1

If m is even and $B \subset B_1(0)$ is a smooth area, then

$$\begin{aligned} \begin{aligned} \int _{B}u^{m^*-1}\xi \frac{\partial K(|y|)}{\partial y_s}&= \int _{\partial B}K(|y|)u^{m^*-1}\xi \nu _s -\int _{\partial B} \Delta ^{\frac{m}{2}}u\Delta ^{\frac{m}{2}}\xi \nu _s\\&\quad +\sum _{i=1}^{\frac{m}{2}}\bigg ( \int _{\partial B}\Delta ^{m-i}u \frac{\partial ^2\Delta ^{i-1}\xi }{\partial y_s\partial \nu }-\int _{\partial B}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\frac{\partial \Delta ^{i-1}\xi }{\partial y_s}\\&\quad +\int _{\partial B} \Delta ^{m-i}\xi \frac{\partial ^2 \Delta ^{i-1}u }{\partial y_s\partial \nu }-\int _{\partial B} \frac{\partial \Delta ^{m-i}\xi }{\partial \nu }\frac{\partial \Delta ^{i-1}u}{\partial y_s}\bigg ), \end{aligned} \end{aligned}$$

(B.3)

and

$$\begin{aligned} \begin{aligned}&\int _{B}u^{m^*-1}\xi \langle \nabla K(|y|),y-x_{0}\rangle \\&\quad = \int _{\partial B}K(|y|)u^{m^*-1}\xi \langle \nu ,y-x_{0}\rangle - \int _{\partial B}\Delta ^{\frac{m}{2}}u\Delta ^{\frac{m}{2}}\xi \langle \nu ,y-x_{0}\rangle \\&\qquad -\sum _{i=1}^{\frac{m}{2}}\bigg (\int _{\partial B}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}\langle \nabla \xi ,y-x_{0}\rangle +\int _{\partial B}\Delta ^{m-i}u\frac{\partial \Delta ^{i-1}\langle \nabla \xi ,y-x_{0}\rangle }{\partial \nu }\\&\qquad -\int _{\partial B} \frac{ \partial \Delta ^{m-i}\xi }{\partial \nu }\Delta ^{i-1}\langle \nabla u,y-x_{0}\rangle +\int _{\partial B} \Delta ^{m-i}\xi \frac{\partial \Delta ^{i-1}\langle \nabla u,y-x_{0}\rangle }{\partial \nu }\bigg )\\&\qquad +\frac{N-2m}{2}\sum _{i=1}^{\frac{m}{2}}\bigg (\int _{\partial B} \Delta ^{m-i}u\frac{\partial \Delta ^{i-1}\xi }{\partial \nu }+\int _{\partial B}\Delta ^{m-i}\xi \frac{\partial \Delta ^{i-1}u}{\partial \nu }\\&\qquad -\int _{\partial B} \frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}\xi -\int _{\partial B}\frac{\partial \Delta ^{m-i}\xi }{\partial \nu }\Delta ^{i-1}u\bigg ), \end{aligned}\nonumber \\ \end{aligned}$$

(B.4)

where $x_0\in B_1(0)$, $\nu$ is the out-of-unit normal derivative of B, $s=1,\ldots ,N$ and $\nu _s$ is the s-th component of $\nu$.

Lemma B.2

If m is odd and $B \subset B_1(0)$ is a smooth area, then

$$\begin{aligned} \begin{aligned}&\int _{B}u^{m^*-1}\xi \frac{\partial K(|y|)}{\partial y_s}\\&\quad = \int _{\partial B}K(|y|)u^{m^*-1}\xi \nu _s -\int _{\partial B} \langle \nabla \Delta ^{\frac{m-1}{2}}u,\nabla \Delta ^{\frac{m-1}{2}}\xi \rangle \nu _s \\&\qquad -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B}\Delta ^{m-i}u \frac{\partial ^2\Delta ^{i-1}\xi }{\partial y_s\partial \nu }+\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\frac{\partial \Delta ^{i-1}\xi }{\partial y_s}\\&\qquad -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B} \Delta ^{m-i}\xi \frac{\partial ^2 \Delta ^{i-1}u }{\partial y_s\partial \nu }+\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B} \frac{\partial \Delta ^{m-i}\xi }{\partial \nu }\frac{\partial \Delta ^{i-1}u}{\partial y_s},\\ \end{aligned} \end{aligned}$$

(B.5)

and

$$\begin{aligned}&\int _{B}u^{m^*-1}\xi \langle \nabla K(|y|),y-x_{0}\rangle \\&\quad = \int _{\partial B}K(|y|)u^{m^*-1}\xi \langle \nu ,y-x_{0}\rangle - \int _{\partial B}\langle \nabla \Delta ^{\frac{m-1}{2}}u,\nabla \Delta ^{\frac{m-1}{2}}\xi \rangle \langle \nu , y-x_{0}\rangle \\&\qquad +\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}\langle \nabla \xi ,y-x_{0}\rangle -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B}\Delta ^{m-i}u\frac{\partial \Delta ^{i-1}\langle \nabla \xi ,y-x_{0}\rangle }{\partial \nu }\\&\qquad +\int _{\partial B} \frac{ \partial \Delta ^{m-i}\xi }{\partial \nu }\Delta ^{i-1}\langle \nabla u ,y-x_{0}\rangle -\sum _{i=1}^{\frac{m-1}{2}} \int _{\partial B} \Delta ^{m-i}\xi \frac{\partial \Delta ^{i-1}\langle \nabla u,y-x_{0}\rangle }{\partial \nu }\\&\qquad -\frac{N-2m}{2}\sum _{i=1}^{\frac{m-1}{2}}\bigg (\int _{\partial B} \Delta ^{m-i}u\frac{\partial \Delta ^{i-1}\xi }{\partial \nu }+\int _{\partial B}\Delta ^{m-i}\xi \frac{\partial \Delta ^{i-1}u}{\partial \nu }\bigg )\\&\qquad +\frac{N-2m}{2}\sum _{i=1}^{\frac{m+1}{2}}\bigg ( \int _{\partial B} \frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}\xi -\int _{\partial B}\frac{\partial \Delta ^{m-i}\xi }{\partial \nu }\Delta ^{i-1}u\bigg ),\\ \end{aligned}$$

where $x_0\in B_1(0)$, $\nu$ is the out-of-unit normal derivative of B, $s=1,\ldots ,N$ and $\nu _s$ is the s-th component of $\nu$.

The proof of the above two lemmas are similar. We only give the outline of proofs. For the first lemma, we multiply the two Eqs. (B.1) and (B.2) by $\frac{\partial \xi }{\partial y_s}$ and $\frac{\partial u }{\partial y_s}$ respectively; for the second part, we multiply the two Eqs. (B.1) and (B.2) by $\langle \nabla \xi ,y-x_0\rangle$ and $\langle \nabla u,y-x_0\rangle$ respectively, finally use integral by parts to get the result. For specific details, please refer to [15] Lemma 2.1 and 2.2.

Next, let us discuss the following identities involving Green function, which is very important when proving non-degenerate properties. We add definitions for N is odd:

$$\begin{aligned}&I_{1,2}(u,v,d)= \int _{\partial B_d}K(|y|)u^{m^*-1}v \nu _1 -\int _{\partial B_d} \langle \nabla \Delta ^{\frac{m-1}{2}}u,\nabla \Delta ^{\frac{m-1}{2}}v \rangle \nu _1 \\&\quad -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B_d}\Delta ^{m-i}u \frac{\partial ^2\Delta ^{i-1}v }{\partial y_1\partial \nu }+\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B_d}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\frac{\partial \Delta ^{i-1}v}{\partial y_1}\\&\quad -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B_d} \Delta ^{m-i}v \frac{\partial ^2 \Delta ^{i-1}u }{\partial y_1\partial \nu }+\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B_d} \frac{\partial \Delta ^{m-i}v }{\partial \nu }\frac{\partial \Delta ^{i-1}u}{\partial y_1}, \\&I_{2,2}(u,v,d)= \int _{\partial B_d}K(|y|)u^{m^*-1}v \langle \nu ,y-x_{x_{k_n},1}\rangle \\ {}&\quad - \int _{\partial B_d}\langle \nabla \Delta ^{\frac{m-1}{2}}u,\nabla \Delta ^{\frac{m-1}{2}}v\rangle \langle \nu , y-x_{x_{k_n},1}\rangle \\&\quad +\sum _{i=1}^{\frac{m+1}{2}}\int _{\partial B_d}\frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}\langle \nabla v ,y-x_{x_{k_n},1}\rangle \\ {}&\quad -\sum _{i=1}^{\frac{m-1}{2}}\int _{\partial B_d}\Delta ^{m-i}u\frac{\partial \Delta ^{i-1}\langle \nabla v ,y-x_{x_{k_n},1}\rangle }{\partial \nu }\\&\quad +\int _{\partial B_d} \frac{ \partial \Delta ^{m-i}v }{\partial \nu }\Delta ^{i-1}\langle \nabla u ,y-x_{x_{k_n},1}\rangle \\ {}&\quad -\sum _{i=1}^{\frac{m-1}{2}} \int _{\partial B_d} \Delta ^{m-i}v \frac{\partial \Delta ^{i-1}\langle \nabla u,y-x_{x_{k_n},1}\rangle }{\partial \nu }\\&\quad -\frac{N-2m}{2}\sum _{i=1}^{\frac{m-1}{2}}\bigg (\int _{\partial B_d} \Delta ^{m-i}u\frac{\partial \Delta ^{i-1}v }{\partial \nu }+\int _{\partial B_d}\Delta ^{m-i}v\frac{\partial \Delta ^{i-1}u}{\partial \nu }\bigg )\\&\quad +\frac{N-2m}{2}\sum _{i=1}^{\frac{m+1}{2}}\bigg ( \int _{\partial B_d} \frac{\partial \Delta ^{m-i}u}{\partial \nu }\Delta ^{i-1}v-\int _{\partial B_d}\frac{\partial \Delta ^{m-i}v }{\partial \nu }\Delta ^{i-1}u\bigg ).\\ \end{aligned}$$

Lemma B.3

For any $d\in (0,\frac{\delta }{k_n} )$, where $\delta >0$ is a fixed small constant, we have

$$\begin{aligned}{} & {} I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=I_{1,2}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d) \nonumber \\{} & {} \quad =-2\frac{\partial H}{\partial y_1}(x_{k_n,1},x_{k_n,1}), \end{aligned}$$

(B.6)

$$\begin{aligned}{} & {} I_{1,1}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}G(y,x_{k_n,j}),d) =I_{1,2}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}G(y,x_{k_n,j}),d) \nonumber \\{} & {} \quad =\sum _{j=2}^{k_n}\frac{\partial G}{\partial y_1}(x_{k_n,1},x_{k_n,j}), \end{aligned}$$

(B.7)

$$\begin{aligned}{} & {} I_{2,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=I_{2,2}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)\nonumber \\{} & {} \quad =(N-2m)H(x_{k_n,1},x_{k_n,1}), \end{aligned}$$

(B.8)

$$\begin{aligned}{} & {} I_{2,1}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}G(y,x_{k_n,j}),d)=I_{2,2}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}G(y,x_{k_n,j}),d)\nonumber \\{} & {} \quad =-\frac{N-2m}{2}\sum _{j=2}^{k_n}G(x_{k_n,1},x_{k_n,j}), \end{aligned}$$

(B.9)

$$\begin{aligned}{} & {} I_{2,1}(G(y,x_{k_n,1}),\frac{\partial G}{\partial x_1}(y,x_{k_n,1}),d)=I_{2,2}(G(y,x_{k_n,1}),\frac{\partial G}{\partial x_1}(y,x_{k_n,1}),d)\nonumber \\{} & {} \quad =(N-2m+1)\frac{\partial H}{\partial y_1} (x_{k_n,1},x_{k_n,1}), \end{aligned}$$

(B.10)

$$\begin{aligned}{} & {} I_{2,1}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}(cos\theta _j \frac{\partial G}{\partial x_1}(x_{k_n,1},x_{k_n,j}) +sin\theta _j \frac{\partial G}{\partial x_2}(x_{k_n,1},x_{k_n,j})),d)\nonumber \\{} & {} \quad =I_{2,2}(G(y,x_{k_n,1}),\sum _{j=2}^{k_n}(cos\theta _j \frac{\partial G}{\partial x_1}(x_{k_n,1},x_{k_n,j}) +sin\theta _j \frac{\partial G}{\partial x_2}(x_{k_n,1},x_{k_n,j})),d)\nonumber \\{} & {} \quad =-\frac{N-2m}{2}\sum _{j=2}^{k_n}\frac{\partial G}{\partial y_1}(x_{k_n,1},x_{k_n,j}), \end{aligned}$$

(B.11)

and

$$\begin{aligned} \begin{aligned}&I_{2,1}\left( \sum _{j=2}^{k_n}G(y,x_{k_n,j}\right) ,\frac{\partial G}{\partial x_1}(y,x_{k_n,1}),d)=I_{2,2}\left( \sum _{j=2}^{k_n}G(y,x_{k_n,j}\right) ,\frac{\partial G}{\partial x_1}(y,x_{k_n,1}),d)\\&\quad =-\frac{N-2m+2}{2}\sum _{j=2}^{k_n}\frac{\partial G}{\partial y_1}(x_{k_n,1},x_{k_n,j}). \end{aligned} \end{aligned}$$

(B.12)

Proof

We give the proving process of (B.6), (B.8) and (B.10). The rest can be deduced similarly. In the first place, in the domain $B_d(x_{k_n,1})\backslash B_\epsilon (x_{k_n,1}),0<\epsilon <d$, we have

$$\begin{aligned} \begin{aligned}&I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)-I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),\epsilon )\\&\quad =\int _{B_d(x_{k_n,1})\backslash B_{\epsilon } (x_{k_n,1})}(-\Delta )^mG(y,x_{k_n,1})\frac{\partial G}{\partial y_1}(y,x_{k_n,1})\\&\qquad +(-\Delta )^mG(y,x_{k_n,j})\frac{\partial G}{\partial y_1}(y,x_{k_n,1})\\&\quad =0. \end{aligned} \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=\lim _{\epsilon \rightarrow 0 }I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),\epsilon ), \end{aligned} \end{aligned}$$

while noting that $G(y,x)=\Gamma (y,x)-H(y,x)$, where $\Gamma (y,x)$ is the foundamental solution:

$$\begin{aligned}{} & {} \Gamma (y,x)=\Gamma _{m,N}(y,x)\nonumber \\{} & {} \quad =\frac{1 }{\omega _N 2^{m-1}(N-2)\ldots (N-2m)(m-1)!|y-x|^{N-2m}}, \end{aligned}$$

(B.13)

and H is a regular function. Then by the linear of $I_{i,j},i,j\in \{1,2\}$ and the symmetry of the integration region we have

$$\begin{aligned} \begin{aligned}&I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),\epsilon )\\&\quad = I_{1,1}(\Gamma (y,x_{k_n,1})-H(y,x_{k_n,1}),\Gamma (y,x_{k_n,1})-H(y,x_{k_n,1}),\epsilon )\\&\quad = I_{1,1}(\Gamma (y,x_{k_n,1}),\Gamma (y,x_{k_n,1}),\epsilon )-2I_{1,1}(\Gamma (y,x_{k_n,1}),H(y,x_{k_n,1}),\epsilon )\\&\qquad +I_{1,1}(H(y,x_{k_n,1}),H(y,x_{k_n,1}),\epsilon )\\&\quad =-2 \frac{\partial H }{\partial y_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1). \end{aligned} \end{aligned}$$

Let $\epsilon \rightarrow 0$, $I_{1,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=-2 \frac{\partial H }{\partial y_1}(x_{k_n,1},x_{k_n,1})$,by the exactly same way we can have $I_{1,2}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=-2 \frac{\partial H }{\partial y_1}(x_{k_n,1},x_{k_n,1})$ so (B.6) is proved.

We start to prove (B.8), a direct calulation leads to

$$\begin{aligned} \begin{aligned}&I_{2,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)-G(y,x_{k_n,1}),G(y,x_{k_n,1}),\epsilon )\\&\quad =\int _{B_d(x_{k_n,1})\backslash B_{\epsilon } (x_{k_n,1})} (\Delta )^mG(y,x_{k_n,1})\langle \nabla G(y,x_{k_n,1}),y-x_{k_n,1}\rangle \\&\qquad +\int _{B_d(x_{k_n,1})\backslash B_{\epsilon } (x_{k_n,1})} (\Delta )^mG(y,x_{k_n,j})\langle \nabla G(y,x_{k_n,1}),y-x_{k_n,1}\rangle \\&\qquad +(N-2m)\int _{B_d(x_{k_n,1})\backslash B_{\epsilon } (x_{k_n,1})}(\Delta )^mG(y,x_{k_n,1})G(y,x_{k_n,1})\\&\quad =0. \end{aligned} \end{aligned}$$

We still have

$$\begin{aligned} I_{2,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=\lim _{\epsilon \rightarrow 0 }I_{2,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),\epsilon ), \end{aligned}$$

using the same calculation method,

$$\begin{aligned} I_{2,1}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=(N-2m)H(x_{k_n,1},x_{k_n,1}). \end{aligned}$$

As for the $I_{2,2}(G(y,x_{k_n,1}),G(y,x_{k_n,1}),d)=(N-2m)H(x_{k_n,1},x_{k_n,1})$ is directly availble. In the following, we give a brief calculation process of (B.10). For $I_{2,1}$: by (B.13) and the symmetry of the area $B_d(x)$ and the regularity of H we have:

$$\begin{aligned} \begin{aligned}&I_{2,1}(G(y,x_{k_n,1}), \frac{\partial G}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = I_{2,1}(\Gamma (y,x_{k_n,1})-H(y,x_{k_n,1}), \frac{\partial (\Gamma -H) }{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = I_{2,1}(\Gamma (y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )-I_{2,1}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\qquad -I_{2,1}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )+I_{2,1}(H(y,x_{k_n,1}), \frac{\partial H }{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = -I_{2,1}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )-I_{2,1}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )+o_{\epsilon }(1), \end{aligned} \end{aligned}$$

By direct computations,

$$\begin{aligned} \begin{aligned}&I_{2,1}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = -\frac{N-2m}{2}\int _{\partial B_\epsilon (x_{k_n,1})}\frac{\partial \Delta ^{m-1}\Gamma }{\partial \nu }\frac{\partial H}{\partial x_1}(y,x_{k_n,1})+o_{\epsilon }(1)\\&\quad = -\frac{N-2m}{2} \frac{\partial H}{\partial x_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1),\\ \end{aligned} \end{aligned}$$

since for m is even we have

$$\begin{aligned} \int _{\partial B_\epsilon (x_{k_n,1})}\frac{\partial \Delta ^{m-1}\Gamma }{\partial \nu }=1, \end{aligned}$$

For the second one:

$$\begin{aligned} \begin{aligned}&I_{2,1}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1}(y,x_{k_n,1}),\epsilon )\\&\quad =-\int _{B_{\epsilon }(x_{k_n,1})}\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu }\langle \nabla H,y-x_{k_n,1}\rangle \\&\qquad + \int _{B_{\epsilon }(x_{k_n,1})} \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})\frac{\partial \langle \nabla H,y-x_{k_n,1}\rangle }{\partial \nu }\\&\qquad +\frac{N-2m}{2}\int _{B_{\epsilon }(x_{k_n,1})}\Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})\frac{\partial H}{\partial \nu }-\frac{N-2m}{2}\\&\qquad \int _{B_{\epsilon }(x_{k_n,1})}\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu } H +o_{\epsilon }(1), \end{aligned} \end{aligned}$$

we will need the following basic results:

$$\begin{aligned} \begin{aligned} \Delta ^{m-1}\Gamma =&\frac{(-1)^{m-1}}{(N-2)w_N|y-x|^{N-2}},\quad \int _{\partial B_\epsilon (x) }\frac{\partial \Delta ^{m-1} \Gamma }{\partial \nu }=(-1)^{m}, \\&\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu }=(-1)^m\frac{(N-1)(y-x_{k_n,1})_1 }{w_N|y-x_{k_n,1}|^{N+1}}. \end{aligned} \end{aligned}$$

A direct calculation shows

$$\begin{aligned}{} & {} -\int _{B_{\epsilon }(x_{k_n,1})}\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu }\langle \nabla H,y-x_{k_n,1}\rangle =-\frac{N-1}{N}\frac{\partial H}{\partial y_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1), \\{} & {} \int _{B_{\epsilon }(x_{k_n,1})} \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})\frac{\partial \langle \nabla H,y-x_{k_n,1}\rangle }{\partial \nu }=\\{} & {} \quad -\int _{B_{\epsilon }(x_{k_n,1})}\frac{(y-x_{k_n,1})_1^2}{|y-x_{k_n,1}|^2}\frac{\partial H}{\partial y_1}(y,x_{k_n,1})+o_{\epsilon }(1) \\{} & {} \quad = -\frac{1}{N}\frac{\partial H}{\partial y_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1). \end{aligned}$$

therefore we have

$$\begin{aligned} \begin{aligned}&I_{2,1}(G(y,x_{k_n,1}), \frac{\partial G}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad =\left( \frac{N-2m}{2}+\frac{N-1}{N}+\frac{1}{N} + \frac{N-2m}{2} \left( \frac{1}{N}+\frac{N-1}{N} \right) \right) \frac{\partial H}{\partial y_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1)\\&\quad =(N-2m+1)\frac{\partial H}{\partial y_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1). \end{aligned} \end{aligned}$$

Let $\epsilon \rightarrow 0$ we get (B.10) for m is even. As for m is odd, similar calculations show that:

$$\begin{aligned}{} & {} I_{2,2}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )=-\frac{N-2m}{2} \frac{\partial H}{\partial x_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1), \\{} & {} I_{2,2}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1}(y,x_{k_n,1}),\epsilon )\\{} & {} \quad =\int _{B_{\epsilon }(x_{k_n,1})}\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu }\langle \nabla H,y-x_{k_n,1}\rangle \\{} & {} \qquad - \int _{B_{\epsilon }(x_{k_n,1})} \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})\frac{\partial \langle \nabla H,y-x_{k_n,1}\rangle }{\partial \nu }\\{} & {} \qquad -\frac{N-2m}{2}\int _{B_{\epsilon }(x_{k_n,1})}\Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})\frac{\partial H}{\partial \nu }+\frac{N-2m}{2}\\{} & {} \qquad \int _{B_{\epsilon }(x_{k_n,1})}\frac{\partial \Delta ^{m-1}(\frac{\partial \Gamma }{\partial x_1})}{\partial \nu } H +o_{\epsilon }(1)\\{} & {} \quad =\left( \frac{1-N}{N}-\frac{N-2m}{2}\frac{1}{N}-\frac{1}{N}+\frac{N-2m}{2}\frac{1-N}{N} \right) \frac{\partial H}{\partial x_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1), \end{aligned}$$

so

$$\begin{aligned} \begin{aligned}&I_{2,2}(G(y,x_{k_n,1}), \frac{\partial G}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = I_{2,2}(\Gamma (y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )-I_{2,2}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\qquad -I_{2,2}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )+I_{2,2}(H(y,x_{k_n,1}), \frac{\partial H }{\partial x_1 }(y,x_{k_n,1}),\epsilon )\\&\quad = -I_{2,2}(\Gamma (y,x_{k_n,1}),\frac{\partial H}{\partial x_1 }(y,x_{k_n,1}),\epsilon )-I_{2,2}(H(y,x_{k_n,1}),\frac{\partial \Gamma }{\partial x_1 }(y,x_{k_n,1}),\epsilon )+o_{\epsilon }(1)\\&\quad =(N-2m+1)\frac{\partial H}{\partial x_1}(x_{k_n,1},x_{k_n,1})+o_{\epsilon }(1). \end{aligned} \end{aligned}$$

(B.10) is proved. $\square$

Appendix C. Green function

In this part, we give the estimate of modified Green function $G_k$, which is used in Lemma 3.1. It’s necessary for the construction of new bubble solutions.

In general, for any function f defined in ${\mathbb {R}}^N$, we define its corresponding function $f^*\in H_s$ as follows. We first define $A_j$ as

$$\begin{aligned} A_j z= \left( r \cos \left( \theta +\frac{2j \pi }{k}\right) , r \sin \left( \theta +\frac{2j \pi }{k}\right) , z''\right) ,\quad j=1, \ldots , k, \end{aligned}$$

where $z= (z', z'')\in {\mathbb {R}}^N$, $z'= (r\cos \theta , r\sin \theta )\in {\mathbb {R}}^2$, $z''\in {\mathbb {R}}^{N-2}$, while

$$\begin{aligned} B_i z= \bigl ( z_1,\ldots , z_{i-1}, -z_i, z_{i+1},\ldots , z_N),\quad i=1, \ldots , N. \end{aligned}$$

Let

$$\begin{aligned} {\bar{f}}(y)=\frac{1}{k}\sum _{j=1}^k f(A_j y), \end{aligned}$$

and

$$\begin{aligned} f^*(y) = \frac{1}{N-1}\sum _{i=2}^N \frac{1}{2} \bigl ( {\bar{f}}(y)+ {\bar{f}}(B_i y)\bigr ). \end{aligned}$$

Then one can easily check that $f^*\in H_s$.

In the following, we discuss the Green’s function of $L_k$. Since $\delta _x$ is not in the space $H_s$, we consider

$$\begin{aligned} L_k u = \delta _x^*, \hbox { in } B_{1}(0), \quad u\in H_s\cap \mathscr {D}_0^{m,2}(B_1(0)). \end{aligned}$$

(C.1)

The solution of (C.1) is denoted as $G_k(y, x)$, which we call it the Green function of $L_k.$ Let

$$\begin{aligned} \delta _x^* =\frac{1}{N-1}\sum _{i=2}^N \frac{1}{2} \Bigl ( \frac{1}{k}\sum _{j=1}^k \delta _{A_j x}+ \frac{1}{k}\sum _{j=1}^k \delta _{B_i A_j x}\Bigr ). \end{aligned}$$

We have

Proposition C.1

The solution $G_k(y, x)$ satisfies

$$\begin{aligned} |G_k(y, x)|\le \frac{C}{N-2m+1}\sum _{i=2}^N \Bigl ( \frac{1}{k}\sum _{j=1}^k \frac{C}{|y- A_j x|^{N-2m}}+ \frac{1}{k}\sum _{j=1}^k \frac{C}{|y- B_i A_j x|^{N-2m}}\Bigr ). \end{aligned}$$

Proof

Let $v_1= G(x,y)$ be the Green’s function of $(-\Delta )^{m}$ in $B_{1}(0)$ with Dirichlet boundary condition, which can constructed the solution of following polyharmonic equation

$$\begin{aligned} \left\{ \begin{aligned}&(-\Delta )^mu= f, \text { in } B_1(0)\\&D^\alpha u|_{\partial \Omega }=0, \text { for }|\alpha |\le m-1. \end{aligned} \right. \end{aligned}$$

(C.2)

f is a datum in a suitable functional space and u is the unknown solution, then

$$\begin{aligned} u(x)=\int _{B_1(0)}G(x,y)f(y)dy,\quad x\in \Omega , \end{aligned}$$

holds ture. As before we denote $[xy]=\bigg ||x|y-\frac{x}{|x|}\bigg |$ for domain $B_1(0)$, we have

$$\begin{aligned}{} & {} G(x,y)=k_{m,N}|x-y|^{2m-N}\int _1^{\frac{[xy]}{|x-y|}}(v^2-1)^{m-1}v^{1-N}dv, \\{} & {} G(x,y)\simeq |x-y|^{2m-N}\min \bigg \{1,\frac{d(x)^md(y)^m}{|x-y|^{2m}}\bigg \},\text { where }d(x)=dist(x,\partial B_1(0)). \end{aligned}$$

Let $v_2$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{m} v= (m^*-1)K(y) u_k^{m^*-2} v_1,&{} \text {in}\; B_{1}(0),\\ D^\alpha v|_{\partial \Omega }=0 \text { for }|\alpha |\le m-1. \end{array}\right. } \end{aligned}$$

Then $v_2\ge 0$ and

$$\begin{aligned}{} & {} v_2(y)= \displaystyle \int _{B_{1}(0)}G(z, y) (m^*-1) u_k^{m^*-2} K(y)v_1\\{} & {} \quad \le C \displaystyle \int _{B_1(0)}\frac{1}{|y-z|^{N-2m}}\frac{1}{|z-x|^{N-2m}}\,dz. \end{aligned}$$

We can continue this process to find $v_i$, which is the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{m} v= (m^*-1) u_k^{m^*-2}K(y) v_{i-1},&{} \text {in}\; B_{1}(0),\\ D^\alpha v|_{\partial \Omega }=0 \text { for }|\alpha |\le m-1. \end{array}\right. } \end{aligned}$$

And satisfies

$$\begin{aligned} \begin{aligned} 0\le&v_i (y)\\ =&\int _{B_{1}(0)}G(z, y) (m^*-1) u_k^{m^*-2} K(y)v_{i-1}\\ \le&C \int _{B_{1}(0)}\frac{1}{|y-z|^{N-2m}}\frac{1}{|z-x|^{N-2m(i-1)}}\,dz\\ \le&\frac{C}{|y-x|^{N-2mi}}. \end{aligned} \end{aligned}$$

Let i be large so that $v_i\in L^\infty (B_{1}(0))$. Define

$$\begin{aligned} v=\sum _{l=1}^i v_l\quad \hbox { and } w= G_k(y, x)- v^*, \end{aligned}$$

We then have

$$\begin{aligned} {\left\{ \begin{array}{ll} L_k w = f,&{}\hbox {in } B_{1}(0),\\ w = 0,&{}\hbox {on } \partial B_{1}(0), \end{array}\right. } \end{aligned}$$

(C.3)

where $f\in L^\infty \cap H_s$. By Theorem 1.1, (C.3) has a solution $w\in H_s\cap \mathscr {D}_0^{m,2}(B_1(0))$.

By the regularity results of polyharmonic Dirichlet boundary conditions (Theorem 2.20 of [17] ), we have

$$\begin{aligned} ||w||_{L^{\infty }(B_1(0))}\le C||f||_{L^{\infty }(B_1(0))}. \end{aligned}$$

Thus the conclusion is proved. $\square$