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

TL;DR

This paper investigates the local behavior of nonnegative solutions to fractional semi-linear elliptic equations with isolated singularities, demonstrating asymptotic radial symmetry for solutions when the fractional order is between 0 and 1.

Contribution

It extends the understanding of solution symmetry for fractional elliptic equations with isolated singularities, generalizing known results for the case when .

Findings

Solutions are asymptotically radially symmetric near singularities

Generalizes classical symmetry results to fractional orders 

Provides a framework for analyzing fractional semi-linear equations

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.