Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in [formula omitted

Abstract

We consider the following variational inclusions system of the form − △ u + u ∈ ∂ 1 F ( u , v ) in R N , − △ v + v ∈ ∂ 2 F ( u , v ) in R N , with u , v ∈ H 1 ( R N ) , where F : R 2 → R is a locally Lipschitz function and ∂ i F ( u , v ) ( i ∈ { 1 , 2 } ) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H 1 ( R N ) × H 1 ( R N ) . The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.