On a nonlinear eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces

TL;DR

This paper investigates a nonlinear eigenvalue problem involving generalized Laplacian operators within Orlicz-Sobolev spaces, establishing the existence, isolation, and properties of eigenvalues without requiring the $ riangle_2$-condition.

Contribution

It introduces a framework for analyzing eigenvalues of elliptic equations with general operators in Orlicz-Sobolev spaces, proving eigenvalue existence and isolation without the $ riangle_2$-condition.

Findings

Existence of two positive constants $\lambda_0$ and $\lambda_1$ with $\lambda_1$ as an eigenvalue.

Eigenvalues below $\lambda_0$ do not exist.

Eigenvalue $\lambda_1$ is isolated on the right.

Abstract

We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two positive constants $\lambda_{0}$ and $\lambda_{1}$ with $\lambda_{0}\leq\lambda_{1}$ such that $\lambda_1$ is an eigenvalue of the problem while any value $\lambda<\lambda_{0}$ cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of $\lambda_{1}$ on the right side. We emphasize that throughout the paper no $\Delta_2$-condition is needed.