On Nontrivial Solutions of Variational-Hemivariational Inequalities with Slowly Growing Principal Parts

Abstract

This paper is concerned with the inclusion −div(a(|∇\_u\_|)∇\_u\_) + ∂\_u\_ G(x, u) ∋ 0 in Ω, with Dirichlet boundary condition u = 0 on ∂Ω, in the case where the higher order part has slow growth and the lower order part is locally Lipschitz. By using a Mountain Pass theorem for variational-hemivariational inequalities without the Palais–Smale condition in Orlicz–Sobolev spaces, we show the existence of nontrivial solutions of the above inclusion.