Periodic Solution for Higher Order Prescribed Curvature Problem in a Strip

Abstract

In this paper, we first construct one bubbling solution for a higher order prescribed curvature problem with periodic boundary condition in a strip. then, we prove the existence of periodic solution for the higher order prescribed curvature problem in ${\mathbb {R}}^N$, if the prescribed curvature function is periodic. The novelty of the paper is that we obtain the periodic solution directly without involving uniqueness and many complicated estimates. We believe our method will be helpful for other problems related to periodic solution and higher order equations.

Appendix A: Some Essential Estimates

In this section, we give some essential estimates which are independent interesting, which we believe they are useful to other related problems involving polyharmonic equations. We use the same notations as in previous sections.

Lemma A.1

It holds

$$\begin{aligned} 0\le P U_{x,\mu }(y)\le \sum _{j=0}^\infty U_{x_j,\mu }(y). \end{aligned}$$

Proof

We have

$$\begin{aligned} \begin{aligned} P U_{x,\mu }(y)=&\int _{\Omega } G(y, z) U^{m^*-1}_{x,\mu }(z)\,dz\\ =&\sum _{j=0}^\infty \int _{\Omega } \Gamma (y, z+LP_j) U^{m^*-1}_{x,\mu }(z)\,dz\\ \le&\sum _{j=0}^\infty \int _{{\mathbb {R}}^N} \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu }(z)\,dz\\ =&\sum _{j=0}^\infty U_{x,\mu }(y+LP_j). \end{aligned} \end{aligned}$$

$\square$

Let $\varphi _{x,\mu } = U_{x,\mu } - PU_{x,\mu }.$ Then $\varphi _{x,\mu }$ has the following expansion.

Lemma A.2

For $x\in B_1(0)$, it holds

$$\begin{aligned} \varphi _{x,\mu }(y)= & {} - \frac{B}{\mu ^{\frac{N-2m}{2}}}\sum _{j=1}^{\infty } \Gamma (y, x+LP_j) +O\Bigl ( \frac{1}{L^{N-2m} \mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned}$$

(A.1)

$$\begin{aligned} \frac{\partial \varphi _{x,\mu }}{\partial x_h}(y)= & {} - \frac{B }{ \mu ^{\frac{N-2m}{2}}}\sum _{j=1}^{\infty } \frac{\partial \Gamma (y, x+LP_j)}{\partial x_h} +O\Bigl ( \frac{1}{L^{N-2m} \mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned}$$

(A.2)

and

$$\begin{aligned} \begin{aligned} \frac{\partial \varphi _{x,\mu }}{\partial \mu }(y)=\frac{B (N-2m)}{ 2\mu ^{\frac{N-2m+2}{2}}}\sum _{j=1}^{\infty } \Gamma (y, x+LP_j) +\frac{1}{\mu }O\Bigl ( \frac{1}{L^{N-2m} \mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned} \end{aligned}$$

(A.3)

where $B=\displaystyle \int _{{\mathbb {R}}^N} U_{0, 1}^{m^*-1}$.

Proof

We have

$$\begin{aligned} P U_{x,\mu }(y)= \sum _{j=0}^\infty \int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu }(z)\,dz. \end{aligned}$$

(A.4)

For $j=0$, it holds

$$\begin{aligned} \int _{\Omega } \Gamma (y, z) U^{m^*-1}_{x,\mu } =&\int _{{\mathbb {R}}^N} \Gamma (y, z) U^{m^*-1}_{x,\mu }-\int _{{\mathbb {R}}^N\setminus \Omega } \Gamma (y, z) U^{m^*-1}_{x,\mu }\\ =&U_{x,\mu }(y)+O\Bigl ( \int _{{\mathbb {R}}^N\setminus \Omega } \frac{1}{|y-z|^{N-2m}} \frac{1}{ |z|^{N+2m}\mu ^{\frac{N+2m}{2}}}\,dz \Bigr )\\ =&U_{x,\mu }(y)+O\Bigl ( \frac{1}{ L^{N-2m}\mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned}$$

while $j> 0$, we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu }\\&\quad =\int _{B_\delta (x)} \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu }+O\Bigl ( \int _{\Omega \setminus B_\delta (x)} \Gamma (y, z+LP_j) \frac{1}{|z-x|^{N+2m}} \frac{1}{ \mu ^{\frac{N+2m}{2}}} \Bigr )\\&\quad = \frac{1}{\mu ^{\frac{N-2m}{2}}} \int _{B_{\delta \mu }(0)} \Gamma (y,\mu ^{-1}z+x+LP_j) U^{m^*-1}_{0,1} +O\Bigl ( \frac{1}{|LP_j|^{N-2m} \mu ^{\frac{N+2m}{2}} } \Bigr ) \\&\quad = \frac{B \Gamma (y, x+ LP_j)}{ \mu ^{\frac{N-2m}{2}}} +O\Bigl ( \frac{1}{|LP_j|^{N-2m} \mu ^{\frac{N+2m}{2}} } \Bigr ). \end{aligned} \end{aligned}$$

So we have proved (A.1).

Using (A.4), we have

$$\begin{aligned} \frac{\partial P U_{x,\mu }}{\partial x_h}(y)= (m^*-1) \sum _{j=0}^\infty \int _{\Omega } \Gamma (y, z+LP_j) U^{m^*-2}_{x,\mu }(z)\frac{\partial U_{x,\mu }}{\partial x_h}(z)\,dz. \end{aligned}$$

(A.5)

For $j=0$, it holds

$$\begin{aligned} \begin{aligned}&(m^*-1) \int _{\Omega } \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial x_h}\\&\quad =(m^*-1) \int _{{\mathbb {R}}^N} \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial x_h}-(m^*-1) \int _{{\mathbb {R}}^N\setminus \Omega } \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial x_h}\\&\quad = \int _{{\mathbb {R}}^N} \Gamma (y,z) (-\Delta )^m\left( \frac{\partial U_{x,\mu }}{\partial x_h} \right) + O\Bigl ( \int _{{\mathbb {R}}^N\setminus \Omega } \frac{1}{|y-z|^{N-2m}} \frac{1}{|z-x|^{N+2m+1}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\&\quad = \frac{\partial U_{x,\mu }}{\partial x_h}(y) + O\Bigl ( \frac{1}{ L^{N-2m}\mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned} \end{aligned}$$

while $j> 0$, we have

$$\begin{aligned}&(m^*-1)\int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial x_h}\\&\quad =-(m^*-1)\int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial z_h}\\&\quad =-(m^*-1)\int _{B_\delta (x)} \Gamma (y, z+ LP_j) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial z_h}\\&\qquad +O \Bigl ( \int _{\Omega \setminus B_\delta (x)} \Gamma (y, z+ LP_j) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial z_h} \Bigr )\\&\quad = \int _{B_\delta (x)} \frac{ \partial \Gamma (y,z+LP_j) }{ \partial z_h } U^{m^*-1}_{x,\mu } + O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\&\quad = \frac{1}{\mu ^{\frac{N-2m}{2}}} \int _{B_{\delta \mu }(0)} \frac{ \partial \Gamma (y,\mu ^{-1}z+x+LP_j) }{ \partial x_h } U^{m^*-1}_{0,1} + O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\&\quad = \frac{1}{\mu ^{\frac{N-2m}{2}}} \frac{ \partial \Gamma (y,x+LP_j) }{ \partial x_h }\int _{B_{\delta \mu }(0)} U^{m^*-1}_{0,1} + O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\&\quad = \frac{B}{\mu ^{\frac{N-2m}{2}}} \frac{ \partial \Gamma (y,x+LP_j) }{ \partial x_h } +O \Bigl ( \frac{1}{\mu ^{\frac{N-2m}{2}}} \frac{1}{|LP_j|^{N-2m+1}} \int _{{\mathbb {R}}^N \setminus B_{\delta \mu }(0)} U_{0,1}^{m^*-1} \Bigr )\\&\qquad + O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\&\quad = \frac{B}{\mu ^{\frac{N-2m}{2}}} \frac{ \partial \Gamma (y,x+LP_j) }{ \partial x_h } + O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ). \\ \end{aligned}$$

So we have proved (A.2).

Using (A.4), we also have

$$\begin{aligned} \frac{\partial P U_{x,\mu }}{\partial \mu }(y)= (m^*-1) \sum _{j=0}^\infty \int _{\Omega } \Gamma (y, z+LP_j) U^{m^*-2}_{x,\mu }(z)\frac{\partial U_{x,\mu }}{\partial \mu }(z)\,dz. \end{aligned}$$

(A.6)

For $j=0$, it holds

$$\begin{aligned} \begin{aligned}&(m^*-1) \int _{\Omega } \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial \mu }\\&\quad =(m^*-1) \int _{{\mathbb {R}}^N} \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial \mu }-(m^*-1) \int _{{\mathbb {R}}^N\setminus \Omega } \Gamma (y, z) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial \mu }\\&\quad = \frac{\partial U_{x,\mu }}{\partial \mu }(y) + \frac{1}{\mu } O\Bigl ( \frac{1}{ L^{N-2m}\mu ^{\frac{N+2m}{2}}} \Bigr ), \end{aligned} \end{aligned}$$

while $j > 0$, we have

$$\begin{aligned}{} & {} (m^*-1)\int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-2}_{x,\mu }\frac{\partial U_{x,\mu }}{\partial \mu }\\{} & {} \quad =\frac{\partial }{\partial \mu } \left( \int _{\Omega } \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu } \right) \\{} & {} \quad =\frac{\partial }{\partial \mu } \left( \int _{B_\delta (x)} \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu } +O\Bigl ( \int _{\Omega \setminus B_\delta (x)} \Gamma (y, z+ LP_j) U^{m^*-1}_{x,\mu } \Bigr ) \right) \\{} & {} \quad = \frac{\partial }{\partial \mu } \left( \frac{1}{\mu ^{\frac{N-2m}{2}}} \int _{B_{\delta \mu }(0)} \Gamma (y,\mu ^{-1}z+x+LP_j) U^{m^*-1}_{0,1} \right) + \frac{1}{\mu } O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\{} & {} \quad = \frac{\partial }{\partial \mu } \left( \frac{\Gamma (y,x+LP_j)}{\mu ^{\frac{N-2m}{2}}} \int _{B_{\delta \mu }(0)} U^{m^*-1}_{0,1} \right) + \frac{1}{\mu } O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\{} & {} \quad = \frac{\partial }{\partial \mu } \left( \frac{B\Gamma (y,x+LP_j)}{\mu ^{\frac{N-2m}{2}}} \right) + \frac{1}{\mu } O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ) \\{} & {} \quad = -\frac{(N-2m)B\Gamma (y,x+LP_j)}{2\mu ^{\frac{N-2m+2}{2}}} + \frac{1}{\mu } O \Bigl ( \frac{1}{|LP_j|^{N-2m}} \frac{1}{\mu ^{\frac{N+2m}{2}}} \Bigr ). \end{aligned}$$

So we have proved (A.3). $\square$

To determine $x_{0,L}$ and $\mu _L$ for the bubbling solution of (1.1), we estimate the following quantities

$$\begin{aligned} \int _{\Omega } (-\Delta )^m (PU_{x,\mu } ) \partial _h( PU_{x,\mu }) -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }). \end{aligned}$$

(A.7)

First, we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (-\Delta )^m (PU_{x,\mu } ) \partial _h( PU_{x,\mu }) -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }) \\&\quad = \int _{\Omega } U_{x,\mu }^{m^*-1} \partial _h( PU_{x,\mu }) -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }). \end{aligned} \end{aligned}$$

(A.8)

Using Lemma A.1, we have

$$\begin{aligned} \begin{aligned}&\left| \int _{\Omega \setminus B_1(x) } U_{x,\mu }^{m^*-1} \partial _h( PU_{x,\mu }) -\int _{\Omega \setminus B_1(x)} K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }) \right| \\&\quad \le C \mu ^{\alpha (h)} \left( \int _{\Omega \setminus B_1(x)} U_{x,\mu }^{m^*-1} PU_{x,\mu } + \int _{\Omega \setminus B_1(x)} ( PU_{x,\mu })^{m^*} \right) \\&\quad \le C \mu ^{\alpha (h)} \int _{\Omega \setminus B_1(x)} \left( \sum \limits _{j=0}^\infty U_{x_j,\mu } \right) ^{m^*} \\&\quad \le C \mu ^{\alpha (h)+N} \int _{\Omega \setminus B_1(x)} \left[ \frac{1}{ (1+\mu |y-x|)^{N-2m}} +\sum \limits _{j=1}^\infty \frac{1}{ (1+\mu |y-x-LP_j|)^{N-2m} } \right] ^{m^*} \\&\quad \le C \mu ^{\alpha (h)+N} \int _{\Omega \setminus B_1(x)} \left[ \frac{1}{ (1+\mu |y-x|)^{N-2m}} +\frac{1}{(1+\mu |y - x|)^{\frac{N-2m}{2} +\theta }} \frac{1}{ (\mu L)^{\frac{N-2m}{2}-\theta } } \right] ^{m^*} \\&\quad \le C \mu ^{\alpha (h)-N}, \end{aligned} \end{aligned}$$

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

On the other hand, we write

$$\begin{aligned} \begin{aligned}&\int _{ B_1(x)} U_{x,\mu }^{m^*-1} \partial _h( PU_{x,\mu }) -\int _{B_1(x)} K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }) \\&\quad = \int _{ B_1(x)} \left( U_{x,\mu }^{m^*-1} - (PU_{x,\mu })^{m^*-1} \right) \partial _h( PU_{x,\mu }) \\&\qquad -\int _{B_1(x)} (K(y)-1) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }) \\&\quad := J_1-J_2. \end{aligned} \end{aligned}$$

Note that for $N< 4m, \; (N-2m)(m^*-2)> N,$ while for $N>4\,m, \; (N-2\,m)(m^*-2)<N.$ Then by Lemma A.2, we have

$$\begin{aligned} \begin{aligned} J_1=&\int _{ B_1(x)} \left( U_{x,\mu }^{m^*-1} - (PU_{x,\mu })^{m^*-1} \right) \partial _hU_{x,\mu } - \int _{B_1(x)} \left( U_{x,\mu }^{m^*-1} - (PU_{x,\mu })^{m^*-1} \right) \partial _h\varphi _{x,\mu } \\ =&(m^*-1)\int _{ B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu } \partial _hU_{x,\mu } + \mu ^{\alpha (h)} O\left( \int _{B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu }^2 \right) \\ =&(m^*-1)\int _{ B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu } \partial _hU_{x,\mu } + \mu ^{\alpha (h)}O\left( \frac{1}{\mu ^{N-2m} L^{2N-4m}} \int _{B_1(x)} U_{x,\mu }^{m^*-2} \right) \\ =&(m^*-1)\int _{B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu } \partial _hU_{x,\mu } + \mu ^{\alpha (h)}O\left( \frac{1}{\mu ^{2\beta }} +\frac{1}{\mu ^N} \right) , \\ \end{aligned} \end{aligned}$$

since $\mu \sim L^{\frac{N-2m}{\beta -N+2m}}.$

We also have

$$\begin{aligned} \begin{aligned} J_2=&\int _{B_1(x)} (K(y)-1) ( PU_{x,\mu })^{m^*-1} \partial _hU_{x,\mu } - \int _{B_1(x)} (K(y)-1) ( PU_{x,\mu })^{m^*-1} \partial _h\varphi _{x,\mu } \\ =&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \partial _hU_{x,\mu } + \mu ^{\alpha (h)} O \left( \int _{B_1(x)} |K(y)-1| U_{x,\mu }^{m^*-1} |\varphi _{x,\mu }| \right) \\ =&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \partial _hU_{x,\mu } + \mu ^{\alpha (h)} O \left( \frac{1}{\mu ^{\frac{N-2m}{2}} L^{N-2m} } \int _{B_1(x)} |y|^\beta U_{x,\mu }^{m^*-1} \right) \\ =&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \partial _hU_{x,\mu } + \mu ^{\alpha (h)} O \left( \frac{|x|^\beta }{(\mu L)^{N-2m} } + \frac{1}{\mu ^N } \right) \\ =&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \partial _hU_{x,\mu } + \mu ^{\alpha (h)} O \left( \frac{1}{\mu ^{2\beta }} +\frac{1}{\mu ^N } \right) , \end{aligned} \end{aligned}$$

since $|x|=o\left( \frac{1}{\mu } \right) .$

Thus, we obtain

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (-\Delta )^m (PU_{x,\mu } ) \partial _h( PU_{x,\mu }) -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \partial _h( PU_{x,\mu }) \\&\quad = (m^*-1)\int _{ B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu } \partial _hU_{x,\mu } -\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \partial _hU_{x,\mu } \\&\qquad + \mu ^{\alpha (h)}O\left( \frac{1}{\mu ^{2\beta }} +\frac{1}{\mu ^N} \right) . \end{aligned} \end{aligned}$$

(A.9)

Proposition A.3

For $h=1, \ldots , N$, we have

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (-\Delta )^m (PU_{x,\mu } ) \frac{\partial PU_{x,\mu }}{\partial x_h} -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \frac{\partial PU_{x,\mu }}{\partial x_h} \\&\quad = \frac{B_h \mu x_h}{\mu ^{\beta _h - 1 } } + O \left( \frac{1}{\mu ^\beta L} + \frac{1}{\mu ^{\beta _M+\sigma -1}} + \frac{(\mu x_h)^2}{\mu ^{\beta _h -1}} \right) , \end{aligned} \end{aligned}$$

(A.10)

where $B_h$ is a non-zero constant, $h=1,\ldots , N$.

Proof

For $h=1,\ldots , N$, we have

$$\begin{aligned} \begin{aligned}&(m^*-1)\int _{ B_1(x)} U_{x,\mu }^{m^*-2} \varphi _{x,\mu } \frac{\partial U_{x,\mu }}{\partial x_h} \\&\quad = -\frac{(m^*-1)B}{\mu ^{\frac{N-2m}{2}}} \sum \limits _{j=1}^\infty \int _{B_1(x)} U_{x,\mu }^{m^*-2} \frac{\partial U_{x,\mu }}{\partial x_h} \Gamma (y,x+LP_j)\\&\qquad +O \left( \frac{\mu }{L^{N-2m} \mu ^{\frac{N+2m}{2}}} \int _{B_1(x)} U_{x,\mu }^{m^* -1} \right) \\&\quad = \frac{ B }{ \mu ^{\frac{N-2m}{2}}} \sum \limits _{j=1}^\infty \int _{B_1(x)} \frac{\partial U_{x,\mu }^{m^*-1} }{ \partial y_h} \Gamma (y,x+LP_j) +O \left( \frac{1}{\mu ^{N-1}L^{N-2m}} \right) \\&\quad = -\frac{ B }{ \mu ^{\frac{N-2m}{2}}} \sum \limits _{j=1}^\infty \int _{B_1(x)} U_{x,\mu }^{m^*-1} \frac{ \partial \Gamma (y,x+LP_j)}{\partial y_h} +O \left( \frac{1}{\mu ^{\beta +2m-1}} \right) \\&\quad = -\frac{ B^2 }{ \mu ^{N-2m}L^{N-2m+1}} \sum \limits _{j=1}^\infty \frac{ \partial \Gamma (y,P_j)}{\partial y_h} \Bigg |_{y=0} +O \left( \frac{1}{\mu ^{\beta +2}L^3} + \frac{1}{\mu ^{\beta +2m-1}} \right) \\&\quad = O\left( \frac{1}{\mu ^\beta L} \right) . \end{aligned} \end{aligned}$$

(A.11)

On the other hand,

$$\begin{aligned} \begin{aligned}&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1} \frac{\partial U_{x,\mu }}{\partial x_h} \\&\quad = \int _{B_\delta (x)} \sum \limits _{i=1}^N a_i |y_i|^{\beta _i} U_{x,\mu }^{m^*-1} \frac{\partial U_{x,\mu }}{\partial x_h} +O \left( \frac{1}{\mu ^{\beta _M+\sigma -1}} \right) \\&\quad = - \int _{B_{\delta \mu } (0)} \sum \limits _{i=1}^N a_i | y_i +\mu x_i |^{\beta _i} \frac{1}{\mu ^{\beta _i - 1}} U_{0,1}^{m^*-1} \frac{\partial U_{0,1}}{\partial y_h} + O \left( \frac{1}{\mu ^{\beta _M+\sigma -1}} \right) \\&\quad = - \int _{B_{\delta \mu } (0)} a_h | y_h +\mu x_h |^{\beta _h} \frac{1}{\mu ^{\beta _h - 1}} U_{0,1}^{m^*-1} \frac{\partial U_{0,1}}{\partial y_h} + O \left( \frac{1}{\mu ^{\beta _M+\sigma -1}} \right) \\&\quad = -\frac{a_h \mu x_h}{\mu ^{\beta _h - 1 }} \int _{B_{\delta \mu } (0)} \beta _h |y_h|^{\beta _h - 2} y_h U_{0,1}^{m^* - 1} \frac{\partial U_{0,1}}{\partial y_h} + O \left( \frac{1}{\mu ^{\beta _M+\sigma -1}} + \frac{(\mu x_h)^2}{\mu ^{\beta _h -1}} \right) \\&\quad = - \frac{B_h \mu x_h}{\mu ^{\beta _h - 1 } } + O \left( \frac{1}{\mu ^{\beta _M+\sigma -1}} + \frac{(\mu x_h)^2}{\mu ^{\beta _h -1}} \right) , \end{aligned} \end{aligned}$$

(A.12)

where

$$\begin{aligned} B_h = a_h\int _{ {\mathbb {R}}^N} \beta _h |y_h|^{\beta _h - 2} y_h U_{0,1}^{m^* - 1} \frac{\partial U_{0,1}}{\partial y_h} \end{aligned}$$

is a non-zero constant.

Combining (A.9), (A.11) and (A.12), we have proved Proposition A.3. $\square$

Proposition A.4

It holds

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (-\Delta )^m (PU_{x,\mu } ) \frac{\partial PU_{x,\mu }}{\partial \mu } -\int _{\Omega } K(y) ( PU_{x,\mu })^{m^*-1} \frac{\partial PU_{x,\mu }}{\partial \mu } \\&\quad = -\frac{(m^*-1) B \int _{{\mathbb {R}}^N} U_{0,1}^{m^*-2} \psi _0 }{ \mu ^{N-2m+1} L^{N-2m}} \sum \limits _{j=1}^\infty \Gamma (P_j,0) -\frac{1}{\mu ^{\beta +1}} \sum \limits _{i\in J} a_i \int _{{\mathbb {R}}^N} |y_i|^{\beta } U_{0,1}^{m^*-1} \psi _0 \\&\qquad +o\left( \frac{1}{\mu ^{\beta + 1}} \right) , \end{aligned} \end{aligned}$$

(A.13)

where $J = \{ j \;: \; \beta _j = \beta \}$ and $\psi _0$ is defined in (2.6).

Proof

We have

$$\begin{aligned} \begin{aligned}&(m^*-1) \int _{B_1(x)} U_{x,\mu }^{m^*-2}\varphi _{x,\mu } \frac{\partial U_{x,\mu }}{\partial \mu } \\&\quad = -\frac{(m^*-1) B}{ \mu ^{\frac{N-2m}{2}}} \sum \limits _{j=1}^\infty \int _{B_1(x)} U_{x,\mu }^{m^*-2} \frac{\partial U_{x,\mu }}{\partial \mu } \Gamma (y,x+LP_j)\\&\qquad +O\left( \frac{1}{L^{N-2m} \mu ^{\frac{N+2m}{2}} \mu } \int _{B_1(x)} U_{x,\mu }^{m^*-1} \right) \\&\quad = -\frac{(m^*-1) B}{ \mu ^{N-2m+1}} \sum \limits _{j=1}^\infty \int _{B_{\mu }(0)} U_{0,1}^{m^*-2} \psi _0 \Gamma (\mu ^{-1}y,LP_j) +O\left( \frac{1}{\mu ^{N+1}L^{N-2m}} \right) \\&\quad = -\frac{(m^*-1) B}{ \mu ^{N-2m+1}} \sum \limits _{j=1}^\infty \int _{B_{\mu }(0)} U_{0,1}^{m^*-2} \psi _0 \Gamma (0,LP_j) +O\left( \frac{1}{\mu ^{\beta + 1 +2m}} +\frac{1}{\mu ^{\beta + 3} L^2} \right) \\&\quad = -\frac{(m^*-1) B \int _{{\mathbb {R}}^N} U_{0,1}^{m^*-2} \psi _0 }{ \mu ^{N-2m+1} L^{N-2m}} \sum \limits _{j=1}^\infty \Gamma (P_j,0) +O\left( \frac{1}{\mu ^{\beta + 1 +2m}} +\frac{1}{\mu ^{\beta + 3} L^2} \right) . \end{aligned} \end{aligned}$$

(A.14)

On the other hand, we have

$$\begin{aligned}&\int _{B_1(x)} (K(y)-1) U_{x,\mu }^{m^*-1}\frac{\partial U_{x,\mu }}{\partial \mu } \nonumber \\&\quad = \int _{B_{\delta } (x)} \sum \limits _{i=1}^N a_i |y_i|^{\beta _i} U_{x,\mu }^{m^*-1} \frac{\partial U_{x,\mu }}{\partial \mu } +O\left( \frac{1}{\mu ^{\beta _M+\sigma + 1}} \right) \nonumber \\&\quad = \int _{B_{\delta \mu } (0)} \sum \limits _{i=1}^N a_i |y_i+\mu x_i|^{\beta _i} \frac{1}{\mu ^{\beta _i+1}}U_{0,1}^{m^*-1} \psi _0 +O\left( \frac{1}{\mu ^{\beta _M+\sigma + 1}} \right) \nonumber \\&\quad = \int _{B_{\delta \mu } (0)} \sum \limits _{i=1}^N a_i |y_i|^{\beta _i} \frac{1}{\mu ^{\beta _i+1}}U_{0,1}^{m^*-1} \psi _0 +o\left( \frac{1}{\mu ^{\beta + 1}} \right) \nonumber \\&\quad = \int _{B_{\delta \mu } (0)} \sum \limits _{i\in J} a_i |y_i|^{\beta } \frac{1}{\mu ^{\beta +1}}U_{0,1}^{m^*-1} \psi _0 +o\left( \frac{1}{\mu ^{\beta + 1}} \right) \nonumber \\&\quad = \frac{1}{\mu ^{\beta +1}} \sum \limits _{i\in J} a_i \int _{{\mathbb {R}}^N} |y_i|^{\beta } U_{0,1}^{m^*-1} \psi _0 +o\left( \frac{1}{\mu ^{\beta + 1}} \right) . \end{aligned}$$

(A.15)

Combining (A.9), (A.14) and (A.15), we finish the proof of Proposition A.4. $\square$