Symmetry and nonexistence of positive solutions for fractional Choquard equations

Pei Ma, Xudong Shang and Jihui Zhang

Vol. 304 (2020), No. 1, 143–167

DOI: 10.2140/pjm.2020.304.143

Abstract

This paper is devoted to study the following Choquard equation:

$${\left(-△\right)}^{\alpha ∕2}u=\left(|x{|}^{\beta -n}\ast u^p\right)u^{p-1},u\ge 0,x\in R^n,$$

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 cases respectively.

Keywords

The method of moving planes, fractional Laplacian, Choquard equation

Mathematical Subject Classification 2010

Primary: 35B06, 35B09, 35B50, 35B53, 35R11

Milestones

Received: 7 November 2018

Revised: 27 April 2019

Accepted: 25 May 2019

Published: 18 January 2020