Lower bounds for Orlicz eigenvalues

TL;DR

This paper establishes lower bounds for eigenvalues of a weighted nonlinear g-Laplacian eigenvalue problem, generalizing existing inequalities and considering complex growth conditions via Young functions.

Contribution

It introduces new methods to derive eigenvalue lower bounds for a broad class of nonlinear operators involving Young functions and weights, extending previous results.

Findings

Derived lower bounds for eigenvalues in terms of G, H, w, and eigenfunction normalization.

Generalized inequalities from p-Laplacian eigenvalue literature to more complex nonlinear operators.

Provided new strategies applicable to a wide range of weighted nonlinear eigenvalue problems.

Abstract

In this article we consider the following weighted nonlinear eigenvalue problem for the $g-$Laplacian $$ -\mathop{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb{R}^n, n\geq 1 $$ with Dirichlet boundary conditions. Here $w$ is a suitable weight and $g=G'$ and $h=H'$ are appropriated Young functions satisfying the so called $\Delta'$ condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of $G$, $H$, $w$ and the normalization $\mu$ of the corresponding eigenfunctions. We introduce some new strategies to obtain results that generalize several inequalities from the literature of $p-$Laplacian type eigenvalues.