Existence and stability of solutions of general semilinear elliptic equations with measure data

TL;DR

This paper develops a unified theory for the existence and stability of solutions to general semilinear elliptic equations with measure data, covering Dirichlet and Poisson problems, and proves stability under weak data convergence.

Contribution

It introduces a comprehensive framework for analyzing solutions of elliptic equations with measure data, including stability results, which was not previously established in a unified manner.

Findings

Existence of solutions under measure data

Stability of solutions with respect to weak convergence

Unified treatment of Dirichlet and Poisson problems

Abstract

We study existence and stability for solutions of $Lu+g(x; u) = \omega$ in the closure of open set $\Omega$ where L is a second order elliptic operator, $g$ a Caratheodory function and $\omega$ a measure in $\bar\Omega$. We present a uni ed theory of the Dirichlet problem and the Poisson equation. We prove the stability of the problem with respect to weak convergence of the data.