On variational and quasi-variational inequalities with multivalued lower order terms and convex functionals

In this paper, we consider the existence and some qualitative properties of solutions of variational inequalities of the form: Find  u∈D(J):  〈A(u),v−u〉+〈F(u),v−u〉+J(v)−J(u)≥0,∀v∈X, and of quasi-variational inequalities of the form: Find  u∈D(Ju):  〈A(u),v−u〉+〈F(u),v−u〉+Ju(v)−Ju(u)≥0,∀v∈X, where AA is a second-order elliptic operator of Leray–Lions type, FF is a multivalued lower order term, JJ and JuJu are convex functionals, and JuJu also depends on uu. We concentrate here in noncoercive cases and use sub-supersolution methods to study the existence and enclosure of solutions, and also the existence of extremal solutions between sub and supersolutions.