On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting

TL;DR

This paper investigates a non-homogeneous eigenvalue problem with variable growth and potential within Orlicz-Sobolev spaces, including optimization of the eigenvalue through potential variation in a variable exponent Lebesgue space.

Contribution

It introduces a novel analysis of eigenvalue problems in Orlicz-Sobolev spaces with variable growth and explores potential optimization within this framework.

Findings

Characterization of eigenvalues in Orlicz-Sobolev spaces.

Existence of eigenvalues under variable growth conditions.

Optimization of eigenvalues with respect to potential V.

Abstract

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential $V$. The problem is analyzed in the context of Orlicz-Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential $V$ when $V$ lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.