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) = \| abla u\|_\|u\|_ and_,(u) = [u]_,\|u\|_, % where = (p_1,…,p_n), =(s_1,…,s_n) and \| abla u\|_ = _i=1^n \|u_x_i\|_p_i % is an anisotropic Sobolev norm, [u]_, is a fractional version of the same anisotropic norm, and \|u\|_ =(_(… (_|u|^p_1dx_1)^p_2/p_1 dx_2… )^p_n/p_n-1 dx_n)^1/p_n % 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., number=2, journal=Topological Methods in Nonlinear Analysis, author=Dussel, Ignacio Ceresa and Bonder, Julián Fernández, year=2024, month=Sep., pages=577–596