Radial Symmetry of Solutions to Fully Nonlinear Choquard Equations

Abstract

In this paper, we consider the Choquard equation involving the fully nonlinear nonlocal operator

\{\begin{array}{ll}{F}_{s,m}u(x)+\omega u(x)=C_{n,t}(|x{|}^{2t-n}\ast u^q)u^{q-1},& x\in {\mathbb{R}}^n,\\ u>0,& x\in {\mathbb{R}}^n,\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{cases}F_{s,m}u(x)+\omega u (x)=C_{n,t}(|x|^{2t-n}\ast u^{q})u^{q-1}, & x\in \mathbb{R}^{n},\\ u>0, & x\in \mathbb{R}^{n},\end{cases}$$\end{document}

where 0 < s, t < 1, m > 0, 2 < q < ∞, ω > −m^2s. We establish the symmetry and monotonicity of its positive solutions by using the direct method of moving planes. The key ingredients are the narrow region principle and decay at infinity theorem for the Choquard equation.

Keywords

Method of moving planes / Choquard equation / fully nonlinear nonlocal operator / symmetry / 35R06 / 35B50 / 47G20

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