Existence of eigenvalues for anisotropic and fractional anisotropic problems via Ljusternik-Schnirelmann Theory

Abstract

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients $$ \Q_{\p}(u) = \frac{\|\nabla u\|_{\p}}{\|u\|_{\p}} \quad\text{and}\quad \Q_{\s,\p}(u) = \frac{[u]_{\s,\p}}{\|u\|_{\p}}, $$% where $\p = (p_1,\dots,p_n)$, $\s=(s_1,\dots,s_n)$ and $$ \|\nabla u\|_{\p} = \sum_{i=1}^n \|u_{x_i}\|_{p_i} $$% is an anisotropic Sobolev norm, $[u]_{\s,\p}$ is a fractional version of the same anisotropic norm, and $$ \|u\|_{\p} =\bigg(\int_{\R}\bigg(\dots \bigg(\int_{\R}|u|^{p_1}dx_1\bigg)^{{p_2}/{p_1}}dx_2\dots \bigg)^{p_n/p_{n-1}}dx_n\bigg)^{1/p_n} $$% is an anisotropic Lebesgue norm. Using the Lusternik-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.