一类分数阶薛定谔方程孤立解的对称性研究

Symmetry Result of Solitary Solutions of Fractional Schr9dinger Equations in Annular Domains

在有界环形区域上,研究了一类分数阶薛定谔方程孤立解的对称性问题。首先将分数阶薛定谔方程转化为包含Bessel位势和Riesz位势的积分方程组,然后利用移动平面法和Hardy-Littlewood-Sobolev不等式,证明了当方程边值为常数时,环形区域必为同心球,方程正解是径向对称的,且随着到对称点的距离增大而单调递减。

The aim of this paper is to investigate the symmetry problem of a class of fractional Schr dinger equations in bounded annular domains.The fractional Schr dinger equations will be transformed into a system of integral equations involving Bessel potentials and Riesz potentials.Then via the methods of moving planes and Hardy-Littlewood-Sobolev inequality,this paper proves that the annular domains must be balls with the same center,and provided that the boundary values of these equations are constants,positive solutions of this system must be radially symmetric and decreasing with the distance from the center.