半空间上Choquard方程的Liouville型定理

Liouville-Type Theorem for Choquard Equation in Half-Space

摘要: 本文研究半空间上Choquard方程$\{\begin{array}{l}-\Delta u\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}}{\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\end{array}$应用积分形式的移动平面法，证明了在参数p的一定取值条件下，该方程不存在非平凡正解。

Abstract: This paper investigates the Choquard equation in a half-space setting $\{\begin{array}{l}-\Delta u\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}}{\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\end{array}$. By applying the method of moving planes in integral form, it is demonstrated that under certain conditions on the parameter p, the equation has no nontrivial positive solutions.

1. 引言

Choquard方程首先于1976年被P. Choquard提出，也被称为非线性的Schrodinger-Newton方程[1]，Choquard方程在量子力学、统计物理和凝聚态物理中有着重要的应用[2]-[4]，因此研究Choquard方程解的存在性结果与非存在性结果有着重要的意义。

在过去的几年中，移动平面法对于证明椭圆方程正解的非存在性结果的证明[5]-[7]，起到了很大的作用。传统的移动平面法的工具是椭圆方程的最大值原理，但是由于Choquard方程卷积项的影响，直接应用最大值原理面临极大的困难，因为非局部项的影响以及非线性项仅仅是连续的，因此不能直接应用通常的最大值原理，所以我们用积分不等式代替最大值原理，目前这种方法已经广泛应用于椭圆方程的移动平面法[8] [9]。在最近一篇论文[10]中，给出了全空间Choquard方程

$-\Delta u\left(x\right)={\displaystyle {\int }_{{ℝ}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|x-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(x\right)\right|}^{p-2}u\left(x\right),x\in {ℝ}^N$ (1.1)

非平凡正解的非存在性，其中p满足$1<p<\frac{N+\alpha }{N-2}$ 。从椭圆方程先验界的角度来看，为了用爆破方法得到有界区域上的方程解的先验界，通常需要全空间上的Liouville型定理和半空间上的Liouville型定理，因此本文参考一篇[11]的做法，给全空间增加边界条件，在此条件下得到半空间解的性质，同样给Choquard方程加上边界条件，有

$\{\begin{array}{l}-\Delta u\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}}{\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N,\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N.\end{array}$ (1.2)

接下来我们有定理1.1成立，其中证明过程在2、3章给出。

定理1.1 假定u是方程(1.2)的非负弱解，$\max \left\{p,N-2\right\}<\alpha <N$ ，满足$1<p<\frac{2N-2}{N-2}$ 时，那么有$u\equiv 0$ 。

接下来的章节给出定理1.1证明，我们证明的主要思路是，先利用经过Kelvin变换后方程(1.2)的解在无穷远处衰减的性质，得到解与其对称点的大小关系，这是为了得出强极值原理效果一样的不等式关系。再利用移动平面法，得出解关于平行于$x_N$ 轴的每一个平面都是对称的性质。最后把解代入方程，得到所要证明的结论。

2. 准备工作与积分不等式

如前所述，由于卷积的影响，直接对方程(1.2)应用移动平面法存在较大的困难，为此我们先把方程化为微分方程与积分方程组成的方程组。我们定义

$\{\begin{array}{l}m\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}\overline{x}},\\ n\left(\overline{x}\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}.\end{array}$ (2.1)

那么由(2.1)的定义得方程(1.2)等价于下面的方程

$\{\begin{array}{l}-\Delta u\left(y\right)=m\left(y\right){\left|u\left(y\right)\right|}^{p-2}u\left(y\right),y\in {ℝ}_{+}^N,\\ \frac{\partial u}{\partial \nu }\left(\overline{x},0\right)=n\left(\overline{x}\right){\left|u\left(\overline{x},0\right)\right|}^{p-2}u\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N.\end{array}$ (2.2)

我们最终的目标是方程组(2.2)的每一个解，关于平行于$x_N$ 轴的每一个平面都是对称的，由于我们无法排除卷积项的影响，因此不能直接应用移动平面法。我们对u，m，n做Kelvin变换，利用它们在无穷远点的衰减性，得到经过Kelvin变换后u，m，n的对称性。下面我们先定义u，m，n的Kelvin变换。对任意的$x_p=\left({\overline{x}}_p,0\right)\in \partial {ℝ}_{+}^N$ ，我们定义u，m，n在x_p点的Kelvin变换为

$v\left(x\right)=\frac{1}{{\left|x-x_p\right|}^{N-2}}u\left(\frac{x-x_p}{{\left|x-x_p\right|}^2}+x_p\right),w\left(x\right)=\frac{1}{{\left|x-x_p\right|}^{N-\alpha }}m\left(\frac{x-x_p}{{\left|x-x_p\right|}^2}+x_p\right),z\left(\overline{x}\right)=\frac{1}{{\left|\overline{x}-{\overline{x}}_p\right|}^{N-\alpha }}n\left(\frac{\overline{x}-{\overline{x}}_p}{{\left|\overline{x}-{\overline{x}}_p\right|}^2}+{\overline{x}}_p\right),$

不失一般性地假设x_p = 0。直接计算可知，Kelvin变换以后的函数v，w，z满足下面的方程组

$\{\begin{array}{l}w\left(y\right)={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }{\left|\overline{x}\right|}^{N-2+\alpha }}}\text{d}\overline{x},y\in {ℝ}_{+}^N,\\ z\left(\overline{x}\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|{\left|y\right|}^{N-2}v\left(y\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }{\left|y\right|}^{N+\alpha }}}\text{d}y,\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\backslash \left\{0\right\},\\ -\Delta v\left(y\right)=\frac{1}{{\left|y\right|}^{\alpha +2}}w\left(y\right){\left|{\left|y\right|}^{N-2}v\left(y\right)\right|}^{p-2}{\left|y\right|}^{N-2}v\left(y\right),y\in {ℝ}_{+}^N,\\ \frac{\partial v}{\partial \nu }\left(\overline{x},0\right)=\frac{1}{{\left|\overline{x}\right|}^{\alpha }}z\left(\overline{x}\right){\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^{p-2}{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right),\left(\overline{x},0\right)\in \partial {ℝ}_{+}^N\backslash \left\{0\right\}.\end{array}$ (2.3)

有了Kelvin变换以后，我们再引入移动平面过程中要用到的一些记号。假定$\lambda >0$ ，我们记

${\Sigma }_{\lambda }=\left\{x\in {ℝ}_{+}^N|x_1>\lambda \right\},\partial {\Sigma }_{\lambda }=\left\{x\in \partial {ℝ}_{+}^N|x_1>\lambda \right\},T_{\lambda }=\left\{x\in {ℝ}_{+}^N|x_1=\lambda \right\}.$

对任意的$x\in {\Sigma }_{\lambda }$ ，我们记$x^{\lambda }$ 为$x$ 关于平面$T_{\lambda }$ 的平面对称点，即

$x^{\lambda }=\left(2\lambda -x_1,x_2,\cdots ,x_N\right)$

引理2.1 对任意的$\lambda >0$ ，我们记

$\begin{array}{l}{\Sigma }_{\lambda }^v=\left\{x\in {\Sigma }_{\lambda }|v\left(x\right)>v\left(x^{\lambda }\right)\right\},\\ \partial {\Sigma }_{\lambda }^v=\left\{x\in \partial {\Sigma }_{\lambda }|v\left(x\right)>v\left(x^{\lambda }\right)\right\},\end{array}$

存在一个大于p的数$\beta$ ，那么对任意的$y\in {\Sigma }_{\lambda }$ 以及$\left(\overline{x},0\right)\in \partial {\Sigma }_{\lambda }\backslash \left\{\left(2\lambda ,0,\cdots ,0\right)\right\}$ ，我们有下面的不等式

$w\left(y\right)-w\left(y^{\lambda }\right)\le {\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\frac{1}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}}\frac{1}{{\left|\overline{x}\right|}^{N-2+\alpha -\left(N-2\right)\beta }}{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^{p-\beta }\left[v{\left(\overline{x},0\right)}^{\beta }-v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }\right]\text{d}\overline{x}$ (2.4)

以及

$z\left(\overline{x}\right)-z\left({\overline{x}}^{\lambda }\right)\le {\displaystyle {\int }_{{\Sigma }_{\lambda }^v}\frac{1}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}}\frac{1}{{\left|y\right|}^{N+\alpha -\left(N-2\right)\beta }}{\left|{\left|y\right|}^{N-2}v\left(y\right)\right|}^{p-\beta }\left[v{\left(y\right)}^{\beta }-v{\left(y^{\lambda }\right)}^{\beta }\right]\text{d}y.$ (2.5)

我们只证明不等式(2.4)，不等式(2.5)的证明是类似的。首先，我们有等式

$w\left(y\right)={\displaystyle {\int }_{\partial {\Sigma }_{\lambda }}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }{\left|\overline{x}\right|}^{N-2+\alpha }}\text{d}\overline{x}}+{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }}\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|\left({\overline{x}}^{\lambda },0\right)-y\right|}^{N-\alpha }{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha }}\text{d}\overline{x}}$ (2.6)

以及

$w\left(y^{\lambda }\right)={\displaystyle {\int }_{\partial {\Sigma }_{\lambda }}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y^{\lambda }\right|}^{N-\alpha }{\left|\overline{x}\right|}^{N-2+\alpha }}\text{d}\overline{x}}+{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }}\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|\left({\overline{x}}^{\lambda },0\right)-y^{\lambda }\right|}^{N-\alpha }{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha }}\text{d}\overline{x}}$ (2.7)

成立，由于有等式$\left|\left(\overline{x},0\right)-y^{\lambda }\right|=\left|\left({\overline{x}}^{\lambda },0\right)-y\right|$ 且$\left|\left(\overline{x},0\right)-y\right|=\left|\left({\overline{x}}^{\lambda },0\right)-y^{\lambda }\right|$ 成立，再通过上面两个方程可以得到

$w\left(y\right)-w\left(y^{\lambda }\right)={\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\left(\frac{1}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}-\frac{1}{{\left|\left(\overline{x},0\right)-y^{\lambda }\right|}^{N-\alpha }}\right)}\left(\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\overline{x}\right|}^{N-2+\alpha }}-\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha }}\right)\text{d}\overline{x}.$ (2.8)

一方面，如果$\left(\overline{x},0\right)\in \partial {\Sigma }_{\lambda }^v=\left\{\left(\overline{x},0\right)\in \partial {\Sigma }_{\lambda }|v\left(\overline{x},0\right)>v\left({\overline{x}}^{\lambda },0\right)\right\}$ ，我们有${\left|\overline{x}\right|}^{N-2}>{\left|{\overline{x}}^{\lambda }\right|}^{N-2}$ ，那么我们有${\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)>{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)$ 。于是

$\begin{array}{l}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\overline{x}\right|}^{N-2+\alpha }}-\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha }}\\ \le \frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^{p-\beta }v{\left(\overline{x},0\right)}^{\beta }}{{\left|\overline{x}\right|}^{N-2-\left(N-2\right)\beta }}-\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^{p-\beta }v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }}{{\left|{\overline{x}}^{\lambda }\right|}^{N-2-\left(N-2\right)\beta }}\\ \le \frac{1}{{\left|\overline{x}\right|}^{N-2-\left(N-2\right)\beta }}{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^{p-\beta }\left[v{\left(\overline{x},0\right)}^{\beta }-v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }\right].\end{array}$ (2.9)

另一方面，如果$\left(\overline{x},0\right)\in \partial {\Sigma }_{\lambda }\backslash \partial {\Sigma }_{\lambda }^v$ ，那么我们有

$\begin{array}{c}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left(\overline{x},0\right)\right|}^p}{{\left|\overline{x}\right|}^{N-2+\alpha }}\le \frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|\overline{x}\right|}^{N-2+\alpha }v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }}v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }\\ =\frac{1}{{\left|\overline{x}\right|}^{N-2+\alpha -\beta \left(N-2\right)}}\frac{{\left|{\left|\overline{x}\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|\overline{x}\right|}^{\left(N-2\right)\beta }v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }}v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }\end{array}$

$\begin{array}{l}\le \frac{1}{{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha -\beta \left(N-2\right)}}\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|{\overline{x}}^{\lambda }\right|}^{\left(N-2\right)\beta }v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }}v{\left({\overline{x}}^{\lambda },0\right)}^{\beta }\\ =\frac{{\left|{\left|{\overline{x}}^{\lambda }\right|}^{N-2}v\left({\overline{x}}^{\lambda },0\right)\right|}^p}{{\left|{\overline{x}}^{\lambda }\right|}^{N-2+\alpha }}.\end{array}$ (2.10)

因此，我们由方程(2.8)和不等式(2.9)、(2.10)得到不等式(2.4)。不等式(2.5)的证明与不等式(2.4)的证明类似，因此我们省略。

引理2.2在定理1.1的假设下，对于任意$\lambda >0$ ，函数$v,{\left(v-v^{\lambda }\right)}^{+}\in L^{2^{*}}\left({\Sigma }_{\lambda }\right)\cap L^{\infty }\left({\Sigma }_{\lambda }\right)$ ，其中$2^{*}=\frac{2N}{N-2}$ 为临界Sobolev指数。此外，存在关于$\lambda$ 单调递减的常数$c^{\lambda }$ ，使得下面不等式成立

$\begin{array}{c}{\displaystyle {\int }_{{\Sigma }_{\lambda }}{\left|\nabla {\left(v-v^{\lambda }\right)}^{+}\right|}^2\text{d}x}\le C_{\lambda }[{\Vert w\left(y\right)\Vert }_{L^{\frac{2N}{N-\alpha }}\left({\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2N}}+{\Vert z\left(\overline{x}\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-\alpha }}\left(\partial {\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2\left(N-1\right)}}\\ +{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\beta -1}]{\displaystyle {\int }_{{\Sigma }_{\lambda }}{\left|\nabla {\left(v-v^{\lambda }\right)}^{+}\right|}^2\text{d}x}.\end{array}$ (2.11)

证明：首先，因为$\lambda >0$ ，所以存在$r>0$ ，使得${\Sigma }_{\lambda }\subset {ℝ}_{+}^N\setminus B_r^{+}\left(0\right)$ ，其中$B_r^{+}\left(0\right)=\left\{x\in {ℝ}_{+}^N|\left|x\right|<r\right\}$ 。因此，我们由$v\left(x\right)$ 的定义以及它在无穷远点的衰减性得

$v,{\left(v-v^{\lambda }\right)}^{+}\in L^{2^{*}}\left({\Sigma }_{\lambda }\right)\cap L^{\infty }\left({\Sigma }_{\lambda }\right).$

其次，为了处理$v\left(x\right),w\left(x\right),z\left(\overline{x}\right)$ 在原点的可能奇性，我们需要作适当的截断。为此，我们选取下面的径向截断函数$\eta ={\eta }_{\epsilon }\in C^1\left({ℝ}_{+}^N,\left[0,1\right]\right)$ ，且它满足

${\eta }_{\epsilon }\left(x\right)=\{\begin{array}{ll}1,\hfill & 2\epsilon \le \left|x-p^{\lambda }\right|\le \frac{1}{\epsilon },\hfill \\ 0,\hfill & \left|x-p^{\lambda }\right|<\epsilon ,\left|x-p^{\lambda }\right|>\frac{2}{\epsilon }.\hfill \end{array}$

此外，我们还要求当$\epsilon <\left|x-p^{\lambda }\right|<2\epsilon$ 的时候，$\left|\nabla \eta \right|\le \frac{2}{\epsilon }$ 。当$\frac{1}{\epsilon }<\left|x-p^{\lambda }\right|<\frac{2}{\epsilon }$ 的时候，$\left|\nabla \eta \right|\le 2\epsilon$ 。容易验证，满足上面条件的截断函数是存在的。

有了上面的截断函数以后，我们定义$\phi ={\phi }^{\epsilon }={\eta }_{\epsilon }^2{\left(v-v^{\lambda }\right)}^{+}$ 以及$\psi ={\psi }^{\epsilon }={\eta }_{\epsilon }{\left(v-v^{\lambda }\right)}^{+}$ ，那么我们由截断函数的定义可知，函数$\Phi$ 和$\Psi$ 在${\Sigma }^{\lambda }$ 内是有定义的。同时，通过一个简单的计算可知

${\left|\nabla \psi \right|}^2=\nabla {\left(v-v^{\lambda }\right)}^{+}\nabla \phi +{\left[{\left(v-v^{\lambda }\right)}^{+}\right]}^2{\left|\nabla \eta \right|}^2.$

有了上面的准备工作以后，我们有下面的估计

$\begin{array}{l}{\displaystyle {\int }_{{\Sigma }_{\lambda }\cap \left\{2\epsilon \le \left|x-p^{\lambda }\right|\le \frac{1}{\epsilon }\right\}}{\left|\nabla {\left(v-v^{\lambda }\right)}^{+}\right|}^2\text{d}x}\\ \le {\displaystyle {\int }_{{\Sigma }_{\lambda }^v}{\left|\nabla {\psi }_i\left(x\right)\right|}^2\text{d}x}\le {\displaystyle {\int }_{{\Sigma }_{\lambda }^v}\nabla \left(v-v^{\lambda }\right)\nabla \phi \text{d}x}+{\displaystyle {\int }_{{\Sigma }_{\lambda }^v}{\left[{\left(v-v^{\lambda }\right)}^{+}\right]}^2{\left|\nabla {\eta }_{\epsilon }\right|}^2\text{d}x}\\ ={\displaystyle {\int }_{{\Sigma }_{\lambda }^v}-}\Delta \left(v-v^{\lambda }\right)\phi \left(y\right)\text{d}y+{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\frac{\partial \left(v-v^{\lambda }\right)}{\partial \nu }\phi \left(\overline{x},0\right)\text{d}\overline{x}}+I_{\epsilon },\end{array}$ (2.12)

其中$I_{\epsilon }={\displaystyle {\int }_{{\Sigma }_{\lambda }^v}{\left[{\left(v\left(y\right)-v\left(y^{\lambda }\right)\right)}^{+}\right]}^2}{\left|\nabla {\eta }_{\epsilon }\right|}^2\text{d}y$ 。

我们断言，当$\epsilon \to 0$ 时，$I_{\epsilon }\to 0$ 。我们定义两个区域

$D_{\epsilon }^1=\left\{x\in {\Sigma }_{\lambda }|\epsilon <\left|x-p^{\lambda }\right|<2\epsilon \right\},D_{\epsilon }^2=\left\{x\in {\Sigma }_{\lambda }|\frac{1}{\epsilon }<\left|x-p^{\lambda }\right|<\frac{2}{\epsilon }\right\}.$

那么有

${\displaystyle {\int }_{D_{\epsilon }^1}{\left|\nabla \eta \right|}^N\text{d}x}\le C\frac{1}{{\epsilon }^N}\cdot {\epsilon }^N=C.$

同理，

${\displaystyle {\int }_{D_{\epsilon }^2}{\left|\nabla \eta \right|}^N}\text{d}x\le C{\epsilon }^N\cdot \frac{1}{{\epsilon }^N}=C.$

因此，由Holder不等式和${\left(v-v^{\lambda }\right)}^{+}\in L^{2^{*}}\left({\Sigma }_{\lambda }\right)$ ，我们得到，当$\epsilon \to 0$ 时，

$I_{\epsilon }\le {\left({\displaystyle {\int }_{D_{\epsilon }^1\cup D_{\epsilon }^2}{\left[{\left(w-w^{\lambda }\right)}^{+}\right]}^{2^{*}}\text{d}x}\right)}^{\frac{2}{2^{*}}}\cdot {\left({\displaystyle {\int }_{{\Sigma }_{\lambda }}{\left|\nabla \eta \right|}^N\text{d}x}\right)}^{\frac{2}{N}}\to 0.$

接下来我们先估计不等式(2.12)的第一部分，方程为下式

简化方程，我们记${\left|y\right|}^{N-2}v\left(y\right)$ |为$f\left(y\right)$ ，一方面$w\left(y\right)>w\left(y^{\lambda }\right)$ 时，

$\begin{array}{l}\frac{1}{{\left|y\right|}^{\alpha +2}}w\left(y\right){\left|f\left(y\right)\right|}^{p-2}f\left(y\right)-\frac{1}{{\left|y^{\lambda }\right|}^{\alpha +2}}w\left(y^{\lambda }\right){\left|f\left(y^{\lambda }\right)\right|}^{p-2}f\left(y^{\lambda }\right)\\ \le \frac{1}{{\left|y\right|}^{\alpha +2}}\left[w\left(y\right)-w\left(y^{\lambda }\right)\right]{\left|f\left(y\right)\right|}^{p-2}f\left(y\right)\\ +w\left(y\right)\left[\frac{1}{{\left|y\right|}^{\alpha +2}}{\left|f\left(y\right)\right|}^{p-2}f\left(y\right)-\frac{1}{{\left|y^{\lambda }\right|}^{\alpha +2}}{\left|f\left(y^{\lambda }\right)\right|}^{p-2}f\left(y^{\lambda }\right)\right].\end{array}$

另一方面$w\left(y\right)<w\left(y^{\lambda }\right)$ 时，

$\begin{array}{l}\frac{1}{{\left|y\right|}^{\alpha +2}}w\left(y\right){\left|f\left(y\right)\right|}^{p-2}f\left(y\right)-\frac{1}{{\left|y^{\lambda }\right|}^{\alpha +2}}w\left(y^{\lambda }\right){\left|f\left(y^{\lambda }\right)\right|}^{p-2}f\left(y^{\lambda }\right)\\ \le w\left(y\right)\left[\frac{1}{{\left|y\right|}^{\alpha +2}}{\left|f\left(y\right)\right|}^{p-2}f\left(y\right)-\frac{1}{{\left|y^{\lambda }\right|}^{\alpha +2}}{\left|f\left(y^{\lambda }\right)\right|}^{p-2}f\left(y^{\lambda }\right)\right]\end{array}$

接下来我们对于这两个不等式分别进行估计，由$\beta$ 的取值范围我们得到$\alpha -\left(N-2\right)\left(\beta -1\right)\ge 0$ 且$\alpha -2+\alpha -\left(N-2\right)\beta \ge 0$ ，再由$\left|y\right|$ 的系数和$f\left(y\right)$ 的单调递减性和在无穷远处的衰减性，结合HLS不等式，可以得到一个关于$\lambda$ 递减的常数$C_{\lambda }$ ，使得下面两个估计成立

${\displaystyle {\int }_{{\Sigma }_{\lambda }^v}\left[w\left(y\right)\left(\frac{1}{{\left|y\right|}^{\alpha +2}}{\left|f\left(y\right)\right|}^{p-2}f\left(y\right)-\frac{1}{{\left|y^{\lambda }\right|}^{\alpha +2}}{\left|f\left(y^{\lambda }\right)\right|}^{p-2}f\left(y^{\lambda }\right)\right)\right]}{\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^{+}{\eta }_{\epsilon }^2\text{d}y$

$\begin{array}{l}\le {\displaystyle {\int }_{{\Sigma }_{\lambda }^v}w}\left(y\right)\frac{{\left|f\left(y\right)\right|}^{p-2}f\left(y\right)}{{\left|f\left(y\right)\right|}^{\beta -1}}\frac{{\left|v\left(y\right)\right|}^{\beta -2}}{{\left|y\right|}^{\alpha +2-\left(N-2\right)\left(\beta -1\right)}}{\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^2\text{d}y\\ \le C_{\lambda }{\displaystyle {\int }_{{\Sigma }_{\lambda }^v}w}\left(y\right){\left|v\left(y\right)\right|}^{\beta -2}{\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^2\text{d}y\\ \le C_{\lambda }{\left({\displaystyle {\int }_{{\Sigma }_{\lambda }^v}w}{\left(y\right)}^{\frac{2N}{N-\alpha }}\text{d}y\right)}^{\frac{N-\alpha }{2N}}{\left({\displaystyle {\int }_{{\Sigma }_{\lambda }^v}v}{\left(y\right)}^{\frac{2N}{N-2}}\text{d}y\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2N}}{\left({\displaystyle {\int }_{{\Sigma }_{\lambda }^v}{\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^{\frac{2N}{N-2}}\text{d}y}\right)}^{\frac{N-2}{N}}\end{array}$ (2.13)

代入(2.4)，得到估计

$\begin{array}{l}{\displaystyle {\int }_{{\Sigma }_{\lambda }^v}\frac{1}{{\left|y\right|}^{\alpha +2}}}\left[w\left(y\right)-w\left(y^{\lambda }\right)\right]{\left|f\left(y\right)\right|}^{p-2}f\left(y\right){\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^{+}{\eta }_{\epsilon }^2\text{d}y\\ \le C_{\lambda }{\displaystyle {\int }_{{\Sigma }_{\lambda }^v}\frac{1}{{\left|y\right|}^{\alpha +2-\left(N-2\right)\left(\beta -1\right)}}}\left[w\left(y\right)-w\left(y^{\lambda }\right)\right]v{\left(y\right)}^{\beta -1}\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]\text{d}y\\ \le C_{\lambda }{\Vert v{\left(\overline{x},0\right)}^{\beta -1}\left(v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2+\alpha }}\left(\partial {\Sigma }_{\lambda }^v\right)}^{}{\Vert v{\left(y\right)}^{\beta -1}\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]\Vert }_{L^{\frac{2N}{N+\alpha }}\left({\Sigma }_{\lambda }^v\right)}^{}\\ \le C_{\lambda }{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}(\partial {\Sigma }_{\lambda }^v)}^{\beta -1}{\Vert v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}\times {\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(y\right)-v\left(y^{\lambda }\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{}\end{array}$ (2.14)

接下来我们再估计不等式(2.12)的第二部分

$\begin{array}{l}{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\frac{\partial \left(v-v^{\lambda }\right)}{\partial \nu }\phi \left(\overline{x},0\right)\text{d}\overline{x}}\\ ={\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\left(\frac{1}{{\left|\overline{x}\right|}^{\alpha }}z\left(\overline{x}\right){\left|f\left(\overline{x},0\right)\right|}^{p-2}f\left(\overline{x},0\right)-\frac{1}{{\left|{\overline{x}}^{\lambda }\right|}^{\alpha }}z\left({\overline{x}}^{\lambda }\right){\left|f\left({\overline{x}}^{\lambda },0\right)\right|}^{p-2}f\left({\overline{x}}^{\lambda },0\right)\right)}{\left[v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\right]}^{+}{\eta }_{\epsilon }^2\text{d}\overline{x}.\end{array}$

相同的方法下，我们同样有

$\begin{array}{l}{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}z}\left(\overline{x}\right)\frac{{\left|f\left(\overline{x},0\right)\right|}^{p-2}f\left(\overline{x},0\right)}{{\left|f\left(\overline{x},0\right)\right|}^{\beta -1}}\frac{{\left|v\left(\overline{x},0\right)\right|}^{\beta -2}}{{\left|\overline{x}\right|}^{\alpha -\left(N-2\right)\left(\beta -1\right)}}{\left[v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\right]}^2\text{d}\overline{x}\\ \le C_{\lambda }{\left({\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}z}{\left(\overline{x}\right)}^{\frac{2\left(N-1\right)}{N-\alpha }}\text{d}\overline{x}\right)}^{\frac{N-\alpha }{2\left(N-1\right)}}{\left({\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}v}{\left(y\right)}^{\frac{2\left(N-1\right)}{N-2}}\text{d}\overline{x}\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2\left(N-1\right)}}{\left({\displaystyle {\int }_{{\Sigma }_{\lambda }^v}{\left[v\left(y\right)-v\left(y^{\lambda }\right)\right]}^{\frac{2N}{N-2}}\text{d}y}\right)}^{\frac{N-2}{N}}\end{array}$ (2.15)

和

$\begin{array}{l}{\displaystyle {\int }_{\partial {\Sigma }_{\lambda }^v}\frac{1}{{\left|\overline{x}\right|}^{\alpha }}}{\left|f\left(\overline{x},0\right)\right|}^{p-2}f\left(\overline{x},0\right)\left(z\left(\overline{x}\right)-z\left({\overline{x}}^{\lambda }\right)\right){\left[v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\right]}^{+}{\eta }_{\epsilon }^2\text{d}\overline{x}\\ \le C_{\lambda }{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(\overline{x},0\right)-v\left({\overline{x}}^{\lambda },0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{}\times {\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(y\right)-v\left(y^{\lambda }\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{}\end{array}$ (2.16)

将(2.13)，(2.14)，(2.15)，(2.16)代入(2.12)，由Sobolev不等式及Sobolev迹不等式可得

$\begin{array}{l}{\displaystyle {\int }_{{\Sigma }_{\lambda }}{\left|\nabla {\left(v-v^{\lambda }\right)}^{+}\right|}^2\text{d}x}\le C_{\lambda }[{\Vert w\left(y\right)\Vert }_{L^{\frac{2N}{N-\alpha }}\left({\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2N}}+{\Vert z\left(\overline{x}\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-\alpha }}\left(\partial {\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2\left(N-1\right)}}\\ +{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\beta -1}]{\displaystyle {\int }_{{\Sigma }_{\lambda }}{\left|\nabla {\left(v-v^{\lambda }\right)}^{+}\right|}^2\text{d}x}.\end{array}$ (2.17)

3. 定理1.1的证明

有了上面替代极大值不等式的估计后，我们接下来使用移动平面法。

引理3.1在定理1.1的假设下，存在常数${\lambda }_0>0$ ，使得对任意的$\lambda \ge {\lambda }_0,y\in {\Sigma }_{\lambda },\overline{x}\in \partial {\Sigma }_{\lambda }$ ，都有$w\left(y\right)\le w\left(y^{\lambda }\right),v\left(y\right)\le v\left(y^{\lambda }\right),z\left(\overline{x}\right)\le z\left({\overline{x}}^{\lambda }\right)$ 成立。

证明：我们用引理2.1与引理2.2来证明我们的结果，由衰减性可知，存在充分大的${\lambda }_0>0$ ，使得当$\lambda >{\lambda }_0$ 的时候，下面的不等式成立：

$\begin{array}{l}{C}_{\lambda }[{\Vert w\left(y\right)\Vert }_{L^{\frac{2N}{N-\alpha }}\left({\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2N}}+{\Vert z\left(\overline{x}\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-\alpha }}\left(\partial {\Sigma }_{\lambda }^v\right)}^{}{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2(N-1)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\frac{\left(N-2\right)\left(\beta -2\right)}{2\left(N-1\right)}}\\ +{\Vert v\left(\overline{x},0\right)\Vert }_{L^{\frac{2\left(N-1\right)}{N-2}}\left(\partial {\Sigma }_{\lambda }^v\right)}^{\beta -1}{\Vert v\left(y\right)\Vert }_{L^{\frac{2N}{N-2}}\left({\Sigma }_{\lambda }^v\right)}^{\beta -1}]<\frac{1}{2}.\end{array}$ (3.1)

将这一不等式代入引理2.2就得到，对任意的$y\in {\Sigma }_{\lambda }$ ，都有$v\left(y\right)\le v\left(y^{\lambda }\right)$ 。再把这一结果代入引理2.1就得到$y\in {\Sigma }_{\lambda }$ 以及$\overline{x}\in \partial {\Sigma }_{\lambda }$ ，都有$w\left(y\right)\le w\left(y^{\lambda }\right)$ 以及$z\left(\overline{x}\right)\le z\left({\overline{x}}^{\lambda }\right)$ 。

下面我们从右至左移动平面$T_{{\lambda }_0}$ ，假定这一过程在${\lambda }_1$ 的地方停止。即按照下面的方式定义${\lambda }_1$ ：

${\lambda }_1=\inf \left\{\lambda |w\left(y\right)\le w\left(y^{\lambda }\right),v\left(y\right)\le v\left(y^{\lambda }\right),z\left(\overline{x}\right)\le z\left({\overline{x}}^{\lambda }\right),\forall y\in {\Sigma }_{\lambda },\overline{x}\in \partial {\Sigma }_{\lambda }\right\}.$

引理3.2 如果${\lambda }_1>0$ ，那么有$w\left(y\right)\equiv w\left(y^{{\lambda }_1}\right),v\left(y\right)\equiv v\left(y^{{\lambda }_1}\right)$ 以及$z\left(\overline{x}\right)\equiv z\left({\overline{x}}^{{\lambda }_1}\right)$ 。

证明：我们用反证法来证明这个引理。如果$w\left(y\right)\cancel{\equiv }w\left(y^{{\lambda }_1}\right)$ 或$v\left(y\right)\cancel{\equiv }v\left(y^{{\lambda }_1}\right)$ ，那么我们断言平面$T_{{\lambda }_1}$ 可以向左边再移动一点点，即存在${\delta }_0>0$ ，使得对任意的$\lambda \in \left[{\lambda }_1-{\delta }_0,{\lambda }_1\right],y\in {\Sigma }_{\lambda }$ 以及$\overline{x}\in \partial {\Sigma }_{\lambda }$ ，都有$v\left(y\right)\le v\left(y^{\lambda }\right),w\left(y\right)\le w\left(y^{\lambda }\right)$ 以及$z\left(\overline{x}\right)\le z\left({\overline{x}}^{\lambda }\right)$ ，这与${\lambda }_1$ 的定义矛盾，因此反设不成立，即引理得证。我们由$v\left(y\right),w\left(y\right),z\left(\overline{x}\right)$ 无穷远处的衰减性和引理2.2得到存在$\delta >0$ ，使得$\lambda \in \left[{\lambda }_1-\delta ,{\lambda }_1\right]$ 都有$w\left(y\right)\le w\left(y^{\lambda }\right),v\left(y\right)\le v\left(y^{\lambda }\right),z\left(\overline{x}\right)\le z\left({\overline{x}}^{\lambda }\right)$ 成立。

定理1.1的证明假定u，m，n是方程(2.1)，(2.2)的非负弱解，满足定理1.1中的假设，那么对任意的$p\in ℝ$ ，如果我们记v，w，z是u，m，n在p点的Kelvin变换，那么u，m，n关于任意经过p点且平行于$x_N$ 轴的平面对称。

我们只需证明v，w，z关垂直于$x_1,x_2,\cdots ,x_{N-1}$ 轴上的N−1个平面对称，不失一般性设这个平面为过原点且垂直于$x_1$ 轴，继续引理3.1和引理3.2的步骤，若${\lambda }_1>0$ ，得到对称，若${\lambda }_1<0$ ，从左向右移动这个平面得到${\lambda }_2$ ，若${\lambda }_2<0$ ，由引理3.2可得${\lambda }_2$ 为替换后的${\lambda }_1$ ，得到对称，若${\lambda }_2>0$ ，得到$x$ 关于这个平面对称点$x^0$ 有下面性质$v\left(x\right)\ge v\left(x^0\right),w\left(x\right)\ge w\left(x^0\right),z\left(\overline{x}\right)\ge z\left({\overline{x}}^0\right)$ ，与${\lambda }_1$ 得到的 $v\left(x\right)\le v\left(x^0\right),w\left(x\right)\le w\left(x^0\right),z\left(\overline{x}\right)\le z\left({\overline{x}}^0\right)$ 性质可得对称。

我们得到u，m只依赖于$x_N$ ，n为常数，得到下面方程组

$\{\begin{array}{l}-\frac{{\text{d}}^{2}u\left(y_N\right)}{\text{d}{y}_N^2}={\displaystyle {\int }_{\partial {ℝ}_{+}^N}\frac{{\left|u\left(0\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}}\text{d}\overline{x}{\left|u\left(y^N\right)\right|}^{p-2}u\left(y^N\right),y^N>0,\\ -\frac{\partial u}{\partial x_N}\left(0\right)={\displaystyle {\int }_{{ℝ}_{+}^N}\frac{{\left|u\left(y^N\right)\right|}^p}{{\left|\left(\overline{x},0\right)-y\right|}^{N-\alpha }}\text{d}y}{\left|u\left(0\right)\right|}^{p-2}u\left(0\right).\end{array}$ (3.2)

得到u关于$x_N$ 是凹函数，且u关于$x_N$ 单调递减，u必然为常数c，否则u关于$x_N$ 严格单调递减那么由它的凹性可知，当N充分大的时候，$u\left(N\right)<0$ ，这与u的非负性矛盾，证得$u\equiv c$ ，由方程又可得$c\equiv 0$ ，证明完毕。