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arXiv · 1403.2346

On the uniqueness of solutions of an nonlocal elliptic system

Kelei Wang·Juncheng Wei
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Abstract

We consider the following elliptic system with fractional laplacian $$ -(-Δ)^su=uv^2,\ \ -(-Δ)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-Δ)^s$ 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.

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Kelei Wang, Juncheng Wei. 2014-03-10. On the uniqueness of solutions of an nonlocal elliptic system. https://arxiv.org/abs/1403.2346

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