The sub- and supersolution method for variational–hemivariational inequalities

Abstract

The well-known method of sub- and supersolutions is a powerful tool for proving existence and comparison results for initial and/or boundary value problems of nonlinear ordinary differential equations as well as for nonlinear partial differential equations of elliptic and parabolic type. The main goal of this paper is to extend the idea of the sub- and supersolution method in a natural and systematic way to quasilinear elliptic variational–hemivariational inequalities. Owing to the intrinsic asymmetry of the latter (where the problems are stated as inequalities rather than as equalities) an appropriate generalization of the notion of sub- and supersolutions to variational–hemivariational inequalities is by no means straightforward. The obtained results of this paper complement the development of the sub- and supersolution method for nonsmooth variational problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu.