Infinitely Many Solutions for Perturbed Hemivariational Inequalities

Abstract

We deal with a perturbed eigenvalue Dirichlet-type problem for an elliptic hemivariational inequality involving the p-Laplacian. We show that an appropriate oscillating behaviour of the nonlinear part, even under small perturbations, ensures the existence of infinitely many solutions. The main tool in order to obtain our abstract results is a recent critical-point theorem for nonsmooth functionals.

A Note on Elliptic Equations Involving the Critical Sobolev Exponent

“…Moreover, we recall that Theorem 2 has been used in order to obtain some theoretical contributions on the existence of either three or infinitely many critical points for suitable functionals defined on reflexive Banach spaces; see [2,3]. As consequences of the above-cited results, on the vast literature on the subject, we mention here some recent works [4,[6][7][8][9][10][11][12] on the existence of weak solutions for some different classes of elliptic problems.…”

“…Remark 2. In conclusion, we just mention that the technical approach adopted in this manuscript has been used in different settings in order to obtain existence and multiplicity results for several kinds of differential problems, for instance, by Bonanno, Molica Bisci and Rȃdulescu in [8] for elliptic problems on compact Riemannian manifolds without boundary and by D'Aguì and Molica Bisci for an elliptic Neumann problem involving the p-Laplacian; see [10].…”

Existence result for differential inclusion with p(x)-Laplacian

“…Our starting point for considering problems with p(x)-Laplacian were the papers of Gasiński-Papageorgiou [13,14,15] and Kourogenic-Papageorgiou [18], where the authors deal with the constant exponent problems i.e. when p(x) = p. Moreover, the similar kind of problems were considered in D'Aguì -Bisci [6] and Marano -Bisci -Motreanu [20]. In the first of this paper, the authors deal with a perturbed eigenvalue Dirichlet-type problem for an elliptic hemivariational inequality involving the p-Laplacian.…”