A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

TL;DR

This paper investigates a class of nonlinear eigenvalue problems involving nonhomogeneous differential operators in Orlicz-Sobolev spaces, establishing conditions for continuous eigenvalue spectra using variational methods.

Contribution

It introduces new conditions on the potential and nonlinearity ensuring continuous eigenvalue families for nonhomogeneous operators in Orlicz-Sobolev spaces.

Findings

Established sufficient conditions for continuous eigenvalues

Applied variational methods to nonhomogeneous quasilinear problems

Illustrated results with specific examples of the operator a(t)

Abstract

We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.