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

In this paper we consider a differential inclusion in ℝ^N involving a p(x)-Laplacian of the type {-Δ_p(x)u+e(x)|u| ^p(x)-2u∈∂j(x,u(x)),in ℝ^N,u∈W ^1,p(x)(ℝ^N), where p:ℝ^N→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^-. Finally, we obtain the existence of at least one nontrivial solution, when α^->p⁺.