Exact Asymptotic Behavior of Singular Positive Solutions of Fractional Semi-Linear Elliptic Equations

TL;DR

This paper characterizes the precise asymptotic behavior near singularities of positive solutions to fractional semi-linear elliptic equations with isolated singularities, advancing understanding of their structure in mathematical analysis.

Contribution

It establishes the exact asymptotic behavior of solutions with isolated singularities for a class of fractional semi-linear elliptic equations, filling a gap in the theoretical understanding.

Findings

Derived the exact asymptotic form of solutions near singularities.

Identified the conditions on p for the asymptotic behavior.

Provided rigorous proofs for the asymptotic characterization.

Abstract

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-\Delta)^\sigma u = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $\sigma \in (0, 1)$ and $\frac{n}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$.