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

IGNACIO CERESA DUSSEL; JULIÁN FERNÁNDEZ BÓNDER

TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

JULIUSZ SCHAUDER CTR NONLINEAR STUDIES

Año: 2024

1230-3429

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients $$ Q_{p}(u) = rac{|abla u|_{p}}{|u|_{p}},quadext{and}quad Q_{s,p}(u) = rac{[u]_{s,p}}{|u|_{p}}, $$ where $p = (p_1,dots,p_n)$, $s=(s_1,dots,s_n)$ and $$ |abla u|_{p} = sum_{i=1}^n |u_{x_i}|_{p_i} $$ is an anisotropic Sobolev norm, $[u]_{s,p}$ is a fractional version of the same anisotropic norm, and $$ |u|_{p} =left(int_{R}left(dots left(int_{R}|u|^{p_1}dx_1ight)^{rac{p_2}{p_1}},dx_2dots ight)^{p_n/p_{n-1}}dx_night)^{1/p_n} $$ 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.