Remarks on nonlinear equations with measures

TL;DR

This paper investigates the existence and properties of solutions to nonlinear elliptic equations with measure data, focusing on conditions for existence and behavior when measures are singular or blow up.

Contribution

It provides necessary and sufficient conditions for solutions to nonlinear equations with measure data, extending understanding to cases with singular measures and blow-up behavior.

Findings

Characterization of existence conditions for measure data problems

Analysis of solutions with measures that blow up on subsets

Extension of solution concepts for singular measures

Abstract

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Gw$.