Measure data problems for a class of elliptic equations with mixed absorption-reaction

TL;DR

This paper investigates the existence and extension of nonnegative solutions for a class of elliptic equations with mixed absorption-reaction terms involving measure data, providing conditions for solvability and extension in bounded domains.

Contribution

It introduces new conditions for the existence and extension of solutions to elliptic equations with mixed absorption-reaction terms involving Radon measures.

Findings

Established existence criteria for solutions with measure data.

Provided conditions for extending solutions across compact subsets.

Analyzed the interplay between absorption and reaction terms in elliptic equations.

Abstract

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-\Delta u+u^p-M|\nabla u|^q=\mu$ in a domain $\Omega\subset\BBR^N$ where $\mu$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $\Omega\setminus K$ where $K$ is a compact subset of $\Omega$ can be extended as a solution of the same equation in $\Omega$