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

TL;DR

This paper classifies isolated singularities of positive solutions to a fractional elliptic equation with a Hardy potential in the supercritical case, extending understanding of singular behavior in fractional PDEs.

Contribution

It provides a comprehensive classification of isolated singularities for fractional Lane-Emden equations in the supercritical regime, including new results for the Sobolev supercritical case.

Findings

Classification of isolated singularities established

Analysis based on integral bounds and Hardy operators

Results applicable in Sobolev supercritical regime

Abstract

In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \quad\quad (-\Delta)^s u = |x|^{\theta} u^{p}\quad {\rm in}\ \ B_1\setminus\{0\},\quad u= h\quad{\rm in}\ \ \mathbb{R}^N\setminus B_1,\quad $$ 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 H\"older 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 $h$.