Elliptic problems with growth in nonreflexive Orlicz spaces and with measure or $L^1$ data

TL;DR

This paper studies nonlinear elliptic problems with measure or L^1 data in nonreflexive Orlicz spaces, establishing existence, uniqueness, and regularity of solutions under broad growth conditions.

Contribution

It introduces a framework for solving elliptic problems with measure or L^1 data in nonreflexive Orlicz spaces without specific growth restrictions, proving existence, uniqueness, and regularity results.

Findings

Existence of solutions with measure data and weakly monotone operators.

Uniqueness and regularity of solutions with L^1 data and strongly monotone operators.

Solutions' gradients belong to Orlicz-Marcinkiewicz spaces.

Abstract

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right. \] where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,\xi)$ is a Carath\'eodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces.