Elliptic inequalities with multi-valued operators: Existence, comparison and related variational–hemivariational type inequalities

We study multi-valued elliptic variational inclusions in a bounded domain Ω⊂RNΩ⊂RN of the form u∈K:0∈Au+∂IK(u)+F(u)+FΓ(u),u∈K:0∈Au+∂IK(u)+F(u)+FΓ(u), where AA is a second order quasilinear elliptic operator of Leray–Lions type, KK is a closed convex subset of some Sobolev space, IKIK is the indicator function related to KK, and ∂IK∂IK denoting its subdifferential. The lower order multi-valued operators FF and FΓFΓ are generated by multi-valued, upper semicontinuous functions f:Ω×R→2R∖{0̸}f:Ω×R→2R∖{0̸} and fΓ:Γ×R→2R∖{0̸}fΓ:Γ×R→2R∖{0̸}, respectively, with Γ⊂∂ΩΓ⊂∂Ω. Our main goals are as follows: First we provide an existence theory for the above multi-valued variational inequalities. Second, we establish an enclosure and comparison principle based on appropriately defined sub–supersolutions, and prove the existence of extremal solutions. Third, by means of the sub–supersolution method provided here, we are going to show that rather general classes of variational–hemivariational type inequalities turn out to be only subclasses of the above general multi-valued elliptic variational inequalities, which in a way fills a gap in the current literature where these kind of problems are studied independently. Finally, the existence of extremal solutions will allow us to deal with classes of multi-valued function ff and fΓfΓ that are neither lower nor upper semicontinuous, which in turn will provide a tool to obtain existence results for variational–hemivariational type inequalities whose Clarke’s generalized directional derivative may, in addition, discontinuously depend on the function we are looking for. This paper, though of surveying nature, provides an analytical framework that allows to present in a unifying way and to extend a number of recent results due to the authors.