Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

TL;DR

This paper investigates the existence and regularity of solutions to nonlinear quasilinear elliptic equations within Orlicz-Sobolev spaces, providing new auxiliary properties, multiple results, and illustrative examples.

Contribution

It introduces new auxiliary properties in Orlicz-Sobolev spaces and establishes multiple existence and regularity results for a broad class of elliptic equations.

Findings

Established new auxiliary properties in Orlicz-Sobolev spaces

Proved two existence results for solutions

Proved two regularity results for solutions

Abstract

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on $(x,u,\nabla u)$, and with a convective term $f$. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.