On singular elliptic equations with measure sources

TL;DR

This paper establishes the existence of solutions for a class of singular elliptic equations involving measure sources, extending the understanding of such problems with general measure data and singular nonlinearities.

Contribution

It proves the existence of solutions for singular elliptic problems with measure sources, accommodating a broad class of nonlinearities and measure data.

Findings

Existence of solutions for singular elliptic equations with measure sources.

Extension to general measure data and nonlinearities.

Framework applicable to bounded domains in \\mathbb{R}^N.

Abstract

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$. Here $\gamma > 0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$.