Symmetry and Nonexistence of Positive Solutions for Fractional Choquard Equations

TL;DR

This paper investigates the symmetry and nonexistence of positive solutions for fractional Choquard equations using the method of moving planes, providing new insights into solution properties in critical and subcritical cases.

Contribution

It introduces a direct method of moving planes to establish symmetry and nonexistence results for fractional Choquard equations.

Findings

Positive solutions are symmetric in the critical case.

No positive solutions exist in the subcritical case.

Method of moving planes is effectively applied to fractional equations.

Abstract

This paper is devoted to study the following Choquard equation \begin{eqnarray*}\left\{ \begin{array}{lll} (-\triangle)^{\alpha/2}u=(|x|^{\beta-n}\ast u^p)u^{p-1},~~~&x\in R^n, u\geq0,\,\,&x\in R^n, \end{array} \right. \end{eqnarray*} where $0<\alpha,\beta<2$, $1\leq p<\infty$, and $n\geq2$. Using a direct method of moving planes, we prove the symmetry and nonexistence of positive solutions in the critical and subcritical case respectively.