Isolated singularities for fractional Lane-Emden equations in the Serrin's supercritical case

Abstract

In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E)(-\Delta {)}^{s}u=|x{|}^{\theta }{u}^p\mathrm{in}{B}_1\setminus \{0\},u=h\mathrm{in}{\mathbb{R}}^N\setminus B_1,$$ where $s\in (0,1)$, $\theta >-2s$, $p>\frac{N+\theta }{N-2s}$, $B_1$ is the unit ball centered at the origin of ${\mathbb{R}}^N$ with $N>2s$. $h$ is a nonnegative Hölder continuous function in ${\mathbb{R}}^N\setminus B_1$. Our analysis of isolated singularities of $(E)$ is based on an integral upper bounds and the study of the Poisson problem with the fractional Hardy operators. It is worth noting that our classification of isolated singularity holds in the Sobolev super critical case $p>\frac{N+2s+2\theta }{N-2s}$ for $s\in (0,1]$ under suitable assumption of $h$.

36