Orlicz-Sobolev versus Holder local minimizer and multiplicity results for quasilinear elliptic equations

TL;DR

This paper investigates a boundary value problem with nonhomogeneous principal part, establishing regularity, minimizer comparisons, and multiple solutions using Orlicz-Sobolev spaces and critical point theory.

Contribution

It introduces new results comparing Orlicz-Sobolev and Hölder local minimizers and proves the existence of multiple solutions for quasilinear elliptic equations.

Findings

Established regularity of solutions.

Compared Orlicz-Sobolev and Hölder minimizers.

Proved existence of multiple solutions.

Abstract

We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $\Omega$, u=0, & on $\partial\Omega$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus H\"{o}lder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.