Isolated singularities for elliptic equations with Hardy operator and source nonlinearity

TL;DR

This paper classifies isolated singularities and establishes existence and stability of positive solutions for a class of semi-linear elliptic equations involving Hardy potentials, using a novel distributional approach.

Contribution

It introduces a new distributional framework to analyze Hardy problems and provides a comprehensive classification and stability results for singular solutions.

Findings

Classified isolated singularities for the Hardy elliptic equation.

Proved existence of positive solutions with isolated singularities.

Established stability of these solutions.

Abstract

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad \Omega\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partial\Omega. \end{equation} We classify the isolated singularities and obtain the existence, the stability of positive solutions of (\ref{0}). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.