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

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