Existence of nontrivial solutions for Kirchhoff-type variational inclusion system in RNRN

In this paper, we consider the following Kirchhoff-type variational inclusion system-(a+b∫RN|∇u|2dx)▵u+V(x)u=∂1F(u,v),inRN,-(c+d∫RN|∇v|2dx)▵v+V(x)v=∂2F(u,v),inRN,with u,v∈H1(RN), where N>2N>2 and F:R2→RF:R2→R is a locally Lipschitz function and ∂iF(u,v)(i=1,2) are the partial generalized gradients in the sense of Clarke. Under proper growth conditions on the nonlinearity F, the existence of nontrivial solutions for the above system is established. Our approach is variational based on the theories of nonsmooth analysis.