DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

EXISTENCE AND MULTIPLICITY RESULTS FOR
HEMIVARIATIONAL INEQUALITIES
MICHAEL E. FILIPPAKIS
We prove an existence and a multiplicity result for hemivariational inequalities in which
the potential −j(z,x) is only partially coercive. Our approach is variational based on the
nonsmooth critical point theory, see Chang, Kourogenis, and Papageorgiou and on an
auxiliary result due to Tang and Wu relating uniform coercivity and subadditivity.
Copyright © 2006 Michael E. Filippakis. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we prove an existence theorem and a multiplicity theorem for nonlinear
hemivariational variational inequalities driven by the p-Laplacian. Hemivariational inequalities
are a new type of variational expressions, which arise in theoretical mechanics
and engineering, when one deals with nonsmooth and nonconvex energy functionals.
For concrete applications, we refer to the book of Naniewicz and Panagiotopoulos [25].
Hemivariational inequalities have intrinsic mathematical interest as a new form of variational
expressions. They include as a particular case problems with discontinuities.
In the last decade, hemivariational inequalities have been studied from a mathematical
viewpoint primarily for Dirichlet problems. We refer to the works of Goeleven et al. [13],
Motreanu and Panagiotopoulos [24], Radulescu and Panagiotopoulos [27], Radulescu
[26], and the references therein. Quasilinear Dirichlet problems were studied recently
by Gasiński and Papageorgiou [9–12]. The study of the Neumann problem is lagging behind.
In the past, Neumann problems with a C 1 energy functional (i.e., smooth potential)
were studied by Mawhin et al. [23], Drabek and Tersian [8] (semilinear problems), and
Huang [17], Arcoya and Orsina [2], Hu and Papageorgiou [16]. The semilinear Neumann
problem with a discontinuous forcing term was studied by Costa and Goncalves [7]with
a forcing term independent of the space variable z ∈ Z, bounded and with zero mean
value.
In this paper, we prove an existence and a multiplicity result for hemivariational inequalities
where the potential −j(z,x) is only partially coercive. Our present approach is
variational based on the nonsmooth critical point theory (see Chang [5]andKourogenis
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 413–421