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

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients _(u) = \|∇ u\|_\|u\|_ ,(u) = [u],\|u\|_,% where = (p₁,,pₙ), =(s₁,,sₙ) and \|∇ u\|_ = ∑ᵢ₌₁ⁿ \|uxᵢ\|pᵢ% is an anisotropic Sobolev norm, [u], is a fractional version of the same anisotropic norm, and \|u\|_ =(∫_( (∫_|u|p₁dx₁)^p₂/p₁dx₂ )^pₙ/pₙ₋₁dxₙ)1/pₙ% 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.