On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

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 $\text{sg}\in (0,1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use 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 $\text{sg}=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).