Symmetry of Solutions for a Fractional System

TL;DR

This paper proves symmetry properties of non-negative solutions to a coupled fractional Laplacian system using the method of moving planes, without requiring decay conditions at infinity.

Contribution

It introduces a direct application of the method of moving planes to fractional systems, establishing symmetry without decay assumptions.

Findings

Solutions are symmetric under certain conditions.

Method applies directly to fractional systems.

No decay at infinity needed for symmetry.

Abstract

We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for continuous $f, g$ and $\alpha, \beta \in (0,2)$. Under some natural assumptions on $f$ and $g$, by applying the \emph{method of moving planes} directly to the system, we obtain symmetry on non-negative solutions without any decay assumption on the solutions at infinity.