一类非线性拉普拉斯方程正解的对称性

Symmetries of Positive Solutions to a Class of Nonlinear Laplace Equations

摘要: 在分数阶微分方程领域中，通过利用有关拉普拉斯算子的极大值原理，以及移动平面法研究了一类非线性分数阶拉普拉斯方程${\left(-\Delta \right)}^{\frac{\alpha }{2}}u\left(x\right)+u\left(x\right)={\left|x\right|}^p{u}^q\left(x\right)$ ，$x\in R^n\backslash \overline{{{\rm B}}_1}$ ，其中$0<\alpha <2$ ，$p<0$ ，$q<0$ ， $\overline{{{\rm B}}_1}:=\left\{x\in R^n|\left|x\right|\le 1\right\}$ ，$R^n\backslash \overline{{{\rm B}}_1}:=\left\{x\in R^n|\left|x\right|>1\right\}$ 的正解$u$ 的径向对称性，其中 $u\in C_{loc}^{1,1}\left(R^n\backslash \overline{{{\rm B}}_1}\right)\cap L_{\alpha }\left(R^n\right)$ ，证明了正解$u$ 是关于原点对称的。

Abstract: In the field of fractional differential equations, the radial symmetry of the positive solution u of a class of nonlinear fractional Laplace equations is studied by using the maximum principle of Laplace operators and the moving plane method: ${\left(-\Delta \right)}^{\frac{\alpha }{2}}u\left(x\right)+u\left(x\right)={\left|x\right|}^p{u}^q\left(x\right)$ , $x\in R^n\backslash \overline{{{\rm B}}_1}$ , among $0<\alpha <2$ , $p<0$ , $q<0$ , $\overline{{{\rm B}}_1}:=\left\{x\in R^n|\left|x\right|\le 1\right\}$ , $R^n\backslash \overline{{{\rm B}}_1}:=\left\{x\in R^n|\left|x\right|>1\right\}$ , $u\in C_{loc}^{1,1}\left(R^n\backslash \overline{{{\rm B}}_1}\right)\cap L_{\alpha }\left(R^n\right)$ . It is proved that the positive solution u is symmetric about the origin.