Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition

TL;DR

This paper investigates the eigenvalue problem for the fractional g-Laplacian in Orlicz spaces without the Δ₂ condition, establishing existence, spectrum properties, and applications to nonlinear problems.

Contribution

It proves the existence of eigenvalues and solutions in fractional Orlicz spaces lacking the Δ₂ condition, extending spectral theory in this setting.

Findings

Existence of nontrivial solutions for the eigenvalue problem.

Closedness of the spectrum and eigenvalue properties.

Application to nonlinear eigenvalue problems.

Abstract

We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugated function satisfy the $\Delta_2$ condition. Our main result is the existence of a nontrivial solution to such a problem; this is achieved by first showing that the corresponding minimization problem has a solution and then applying a generalized Lagrange multiplier theorem to get the existence of an eigenvalue. Further, we prove closedness of the spectrum and some properties of the eigenvalues and, as an application, we show existence for a class of nonlinear eigenvalue problems.