#### Vol. 304, No. 1, 2020

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Symmetry and nonexistence of positive solutions for fractional Choquard equations

### Pei Ma, Xudong Shang and Jihui Zhang

Vol. 304 (2020), No. 1, 143–167
##### Abstract

This paper is devoted to study the following Choquard equation:

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

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