Existence of Eigenvalues for Anisotropic and Fractional Anisotropic Problems via Ljusternik-Schnirelmann Theory

TL;DR

This paper proves the existence of eigenvalues for anisotropic and fractional anisotropic problems using Ljusternik-Schnirelmann theory, establishing critical values of specific Rayleigh-type quotients and their associated Euler-Lagrange equations.

Contribution

It introduces a novel application of Ljusternik-Schnirelmann theory to anisotropic and fractional anisotropic eigenvalue problems, identifying critical values and their properties.

Findings

Existence of a sequence of critical values for anisotropic and fractional problems.

Derivation of Euler-Lagrange equations for critical points.

Analysis of the relationship between fractional and local critical values.

Abstract

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients $$Q_{\mathbf p}(u) = \frac{\|\nabla u\|_{\mathbf p}}{\|u\|_{\mathbf p}},\quad\text{and}\quad Q_{\mathbf s,\mathbf p}(u) = \frac{[u]_{\mathbf s,\mathbf p}}{\|u\|_{\mathbf p}}, $$ where $\mathbf p = (p_1,\dots,p_n)$, $\mathbf s=(s_1,\dots,s_n)$ and $$ \|\nabla u\|_{\mathbf p} = \sum_{i=1}^n \|u_{x_i}\|_{p_i} $$ is an anisotropic Sobolev norm, $[u]_{\mathbf s,\mathbf p}$ is a fractional version of the same anisotropic norm, and $\|u\|_{\mathbf p}$ is an anisotropic Lebesgue norm. Using the Ljusternik-Schnirelmann theory, we prove the existence of a sequence of critical values and we also find an associated Euler-Lagrange equation for critical points. Additionally, we analyze the connection between the fractional critical values and its local counterparts.