On isolated singularities of fractional semi-linear elliptic equations

Abstract

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\mathrm{\Delta }{)}^{\sigma }u=u^p$ with an isolated singularity, where $\sigma \in (0,1)$ and $\frac{n}{n-2\sigma }<p<\frac{n+2\sigma }{n-2\sigma }$. We first use the blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sigma =1$, this classification result has been proved by Gidas and Spruck (1981) [23], Caffarelli et al. (1989) [7].