Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without the ${\Delta }_2$ condition

Abstract

In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$(-{\Delta }_m{)}^{s}u(x):=2\text{p.v.}{\int }_{{\mathbb{R}}^n}m\left(\frac{|u(x)-u(y)|}{|x-y{|}^s}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y{|}^{n+s}},$$ This operator serves as a model for nonlocal, nonstandard growth diffusion problems. In contrast to previous analyses, we explore the eigenvalue problem without presuming the ${\Delta }_2$ condition on $M$ -- the primitive function of $m$. Our results show the existence of a sequence of eigenvalues ${\lambda }_k\to \infty$. This research contributes to advancing our understanding of nonlocal diffusion models, specifically those characterized by the fractional $m-$Laplacian, by relaxing the constraints imposed by the ${\Delta }_2$ condition.

arXiv comment29 pages. Submitted