Multi-valued variational inequalities in unbounded domains: existence, comparison and extremal solutions

Abstract

In this paper, we study multi-valued quasilinear elliptic variational inequalities of the form in all of as well as in the exterior domain , where is the p-Laplacian, K is a closed convex subset of the Beppo-Levi space with 1<p<N, or with p = N, and is the indicator functional corresponding to K with its subdifferential . The lower order multi-valued operator is generated by a multi-valued, upper semicontinuous function , and the measurable coefficient a is supposed to decay like: Our main goals are as follows. First, we provide a new approach to an existence theory for the above multi-valued variational inequalities in all for 1<p<N as well as in the exterior domain Ω for the borderline case . 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 show that certain rather general classes of variational-hemivariational type inequalities turn out to be only subclasses of the above class of multi-valued elliptic variational inequalities. Finally, we apply the abstract theory to a multi-valued obstacle problem.