Symmetric Properties for Choquard Equations Involving Fully Nonlinear Nonlocal Operators

Abstract

In this paper we consider the following nonlinear nonlocal Choquard equation

$$\begin{aligned} {{\mathcal {F}}} _{\alpha }\left( u(x)\right) +\omega u(x) = C_{n,2s} \left( |x|^{2s-n}*u^q(x)\right) u^r(x), ~ x\in {\mathbb {R}}^n, \end{aligned}$$

where $0<s<1, ~0< \alpha < 2, {\mathcal {F}_\alpha }$ is the fully nonlinear nonlocal operator:

$$\begin{aligned} {\mathcal {F}_\alpha }(u(x)) = {C_{n,\alpha }}P.V.\int _{{\mathbb {R}^n}} {\frac{{F(u(x) - u(y))}}{{{{\left| {x - y} \right| }^{n + \alpha }}}}dy}. \end{aligned}$$

The positive solution to nonlinear nonlocal Choquard equation is shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. We first turn single equation into equivalent system of equations. Then the key ingredients are to obtain the “narrow region principle” and “decay at infinity” for the corresponding problems. We also get radial symmetry results of positive solution for the Schrödinger-Maxwell nonlocal equation. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.