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

In this work we analyze the eigenvalue problem associated to the fractional $m-$Laplacian, defined as $$(-{\mathrm{\Delta }}_m{)}^{s}u(x):=2\text{p.v.}{\int }_{{\mathbb{R}}^n}m\left(\frac{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^s}\right)\frac{(u(x)-u(y))}{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }}\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+s}},$$ 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 ${\mathrm{\Delta }}_2$ condition on $M$ – the primitive function of $m$. Our results show the existence of a sequence of eigenvalues ${\lambda }_k\to \mathrm{\infty }$. 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 ${\mathrm{\Delta }}_2$ condition.