Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data

TL;DR

This paper investigates the existence and stability of weak solutions to a class of semilinear elliptic equations with Leray-Hardy potential and measure data, highlighting differences based on the measure's concentration.

Contribution

It introduces a capacity framework for solvability conditions and analyzes solutions with measure data near the singularity at the origin.

Findings

Solutions differ depending on whether the measure is diffuse or concentrated at the origin.

Necessary and sufficient conditions for solvability are established for power-type nonlinearities.

The study extends understanding of elliptic equations with singular potentials and measure data.

Abstract

We study existence and stability of solutions of (E 1) --$\Delta$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\Omega$, u = 0 on $\partial$$\Omega$, where $\Omega$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $\nu$ is a Radon measure on $\Omega$. We show that the situation differs according $\nu$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.