Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness

TL;DR

This paper proves existence and uniqueness of solutions for a broad class of elliptic equations with irregular data and singular nonlinearities, including cases with finite or infinite energy solutions, and explores their regularity properties.

Contribution

It introduces new existence and uniqueness results for elliptic problems with measure data and singular nonlinearities, extending previous theories to more general settings.

Findings

Existence and uniqueness of solutions with irregular data

Regularity results depending on data smoothness

Optimality discussion of regularity conditions

Abstract

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -\Delta u= h(u){f} \ \ \text{in}\,\ \Omega, $$ where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.