Semilinear elliptic equations with Dirichlet operator and singular nonlinearities

TL;DR

This paper investigates semilinear elliptic equations involving Dirichlet operators and singular nonlinearities, establishing existence, uniqueness, regularity, and stability of solutions under measure convergence.

Contribution

It introduces a framework for analyzing solutions to elliptic equations with singular nonlinearities involving Dirichlet operators, including stability analysis with respect to measure convergence.

Findings

Existence and uniqueness of solutions established.

Regularity results for solutions proved.

Stability of solutions under measure convergence demonstrated.

Abstract

In the paper we consider elliptic equations of the form $-Au=u^{-\gamma}\cdot\mu$, where $A$ is the operator associated with a regular symmetric Dirichlet form, $\mu$ is a positive nontrivial measure and $\gamma>0$. We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials.