Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations

Abstract

This paper is devoted to study the following Choquard equation \begin{eqnarray*}\left\{ \begin{array}{lll} (-\triangle)^{α/2}u=(|x|^{β-n}\ast u^p)u^{p-1},~~~&x\in R^n, u\geq0,\,\,&x\in R^n, \end{array} \right. \end{eqnarray*} where $0<\alpha ,\beta <2$, $1\le p<\infty$, and $n\ge 2$. Using a direct method of moving planes, we prove the symmetry and nonexistence of positive solutions in the critical and subcritical case respectively.

References in corpus (1)

• Some Liouville theorems for the fractional Laplacian