Infinitely many solutions for an anisotropic differential inclusion on unbounded domains

Abstract

The problem deals with the anisotropic p ( x ) -Laplacian operator where p i are Lipschitz continuous functions 2 ≤ p i ( x ) < N for all x ∈ R N and i ∈ { 1 , … , N } . Assume p N o ( x ) = max 1 ≤ i ≤ N p i ( x ) , a ∈ L + 1 ( R N ) ∩ L N p N o ( x ) − 1 ( R N ) , $F(x,t)$ is locally Lipschitz in the $t$-variable integrand and $\partial F(x,t)$ is the subdifferential with respect to the $t$-variable in the sense of Clarke. By establishing the existence of infinitely many solutions, we achieve a first result within the anisotropic framework.