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 $$Q_{\mathbf{p}}(u)=\frac{\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathbf{p}}}{\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathbf{p}}},\text{and}{Q}_{\mathbf{s},\mathbf{p}}(u)=\frac{[u{]}_{\mathbf{s},\mathbf{p}}}{\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathbf{p}}},$$ where $\mathbf{p}=(p_1,\dots ,p_n)$, $\mathbf{s}=(s_1,\dots ,s_n)$ and $$\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathbf{p}}=\sum _{i=1}^n\mathrm{\parallel }{u}_{x_i}{\mathrm{\parallel }}_{p_i}$$ is an anisotropic Sobolev norm, $[u{]}_{\mathbf{s},\mathbf{p}}$ is a fractional version of the same anisotropic norm, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathbf{p}}$ 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.