On the uniqueness of solutions of an nonlocal elliptic system

Abstract

We consider the following elliptic system with fractional laplacian $$ -(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-\Delta)^s$ is the $s$-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis to $s$-harmonic functions. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.