Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities

Abstract

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-\Delta)^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}$ with an isolated singularity, where $\sigma\in (0,1)$. We prove that all the solutions are asymptotically radially symmetric. When $\sigma =1$, these have been proved in \cite{CGS} by Caffarelli, Gidas and Spruck.