The existence of radial solutions for differential inclusion problems in RNRN involving the p(x)p(x)-Laplacian

In this paper we consider a differential inclusion in RNRN involving a p(x)p(x)-Laplacian of the type equation(P){−Δp(x)u+e(x)|u|p(x)−2u∈∂j(x,u(x)),in RN,u∈W1,p(x)(RN), where p:RN→Rp:RN→R is a continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, on the basis of the Weierstrass Theorem and the Mountain Pass Theorem, we prove that there exist at least two nontrivial solutions, when α+p+α−>p+.