Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in RNRN

We consider the following variational inclusions system of the form−△u+u∈∂1F(u,v)in RN,−△v+v∈∂2F(u,v)in RN, with u,v∈H1(RN)u,v∈H1(RN), where F:R2→RF:R2→R is a locally Lipschitz function and ∂iF(u,v)∂iF(u,v) (i∈{1,2}i∈{1,2}) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity FF 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 H1(RN)×H1(RN)H1(RN)×H1(RN). The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.