Multiplicity of Solutions for a problem in double weighted Orlicz-Sobolev and its spectrum

TL;DR

This paper explores the existence of multiple solutions and eigenvalues for a nonlinear elliptic problem involving a weighted -Laplacian operator, using variational methods and Ljusternik-Schnirelmann theory.

Contribution

It establishes the existence of three weak solutions and analyzes eigenvalues in weighted Orlicz-Sobolev spaces without relying solely on the _2-condition.

Findings

Proves the existence of three weak solutions.

Identifies sequences of variational eigenvalues.

Analyzes eigenvalues with and without the _2-condition.

Abstract

The purpose of this paper is to investigate the existence of three different weak solutions to a nonlinear elliptic problem that is governed by the weighted {\varphi}-Laplacian operator and subjected to Dirichlet boundary conditions. We also examine the presence of sequences of variational eigenvalues in two distinct scenarios, one with and one without assuming the {\Delta_2}-condition. Our main results are obtained through technical proofsthat combine a Lagrange multipliers type approach with the Ljusternik-Schnirelmann argument.