Measure Data for a General Class of Nonlinear Elliptic Problems

TL;DR

This paper investigates measure data problems for a broad class of nonlinear elliptic equations involving maximal monotone graphs and Leray-Lions type operators, using anisotropic Sobolev spaces to define solutions.

Contribution

It introduces a new framework for defining solutions to nonlinear elliptic problems with measure data involving maximal monotone graphs and Leray-Lions operators.

Findings

Established existence of solutions in anisotropic Sobolev spaces.

Extended the notion of solutions to include measure data.

Provided a functional setting for these nonlinear problems.

Abstract

We consider nonlinear elliptic inclusion having a measure in the right-hand side of the type $\beta(u)-div a(x,Du)\ni \mu$ in $\Omega$ a bounded domain in $\mathbb{R}^{N},$ with $\beta$ is a maximal monotone graph in $\mathbb{R}^2$ and $a(x,Du)$ is a Leray-Lions type operator. We study a suitable notion of solution for this kind of problem. The functional setting involves anisotropic Sobolev spaces.