A Lazer-McKenna type problem with measures

TL;DR

This paper investigates a class of singular boundary value problems involving measures, extending classical results to cases where the source term is a measure and the nonlinearity is singular.

Contribution

It introduces existence and uniqueness results for solutions to a measure-driven singular PDE with measure data and nonlinear singularities.

Findings

Established existence of solutions for the problem.

Proved uniqueness under certain conditions.

Extended classical results to measure data cases.

Abstract

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega\,. \end{cases} $$ Here $\mu$ is a nonnegative bounded Radon measure on a bounded open set $\Omega\subset\mathbb{R}^N$, and $\gamma>0$.