Singularities of fractional Emden's equations via Caffarelli-Silvestre extension

TL;DR

This paper investigates isolated singularities of fractional Emden equations using the Caffarelli-Silvestre extension, deriving a priori estimates and analyzing self-similar solutions to characterize singular behaviors.

Contribution

It introduces a novel approach employing the Caffarelli-Silvestre extension to analyze singularities in fractional Emden equations, providing new a priori estimates and solution classifications.

Findings

Characterization of singularities via self-similar solutions

Development of a priori estimates for solutions

Application of energy methods to fractional PDEs

Abstract

We study the isolated singularities of functions satisfying (E) (--$\Delta$) s v$\pm$|v| p--1 v = 0 in $\Omega$\{0}, v = 0 in R N \$\Omega$, where 0 < s < 1, p > 1 and $\Omega$ is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + x R N. We emphasize the obtention of a priori estimates, analyse the set of self-similar solutions via energy methods to characterize the singularities.