Multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach (Q720647)

The author considers the following problem for the unknown function \(u\): \[ \alpha'(\Psi(u))\left[-\text{div}(A(x,u)|\nabla u|^{p(x)-2}\nabla u)+\frac{A'_t(x,u)}{p(x)}|\nabla u|^{p(x)}+|u|^{p(x)-2}u\right]=f(x,u) \] in \(\mathbb{R}^n \), \noindent where \(\Psi(\cdot)\) represents a suitable integral operator, and \(A\) and \(f\) are symmetric in their second entries. This is a quasi-linear nonlocal (Kirchhoff type) problem with symmetry, involving variable exponents. It can be used for describing the motion of a fluid in a non-homogeneous anisotropic medium. The author proves that, under quite general hypotheses, the problem has infinitely many weak solutions. The proof makes use of the nonsmooth critical point theory.