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