Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator

TL;DR

This paper proves symmetry and monotonicity of positive solutions to a class of Choquard equations with fully nonlinear nonlocal operators using the moving plane method, establishing key principles like narrow region and decay at infinity.

Contribution

It introduces a novel application of the moving plane method to fully nonlinear nonlocal Choquard equations, including new principles for analysis.

Findings

Positive solutions are symmetric and monotone.

Established narrow region principle for nonlocal operators.

Proved decay at infinity for solutions.

Abstract

In this paper, the positive solutions to Choquard equation involving fully nonlinear nonlocal operator are shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. The key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.