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In this paper, we study multi-valued quasilinear elliptic variational inequalities of the formin all ofas well as in the exterior domain, whereis the  p -Laplacian,  K  is a closed convex subset of the Beppo-Levi spacewith 1&lt; p &lt; N , orwith  p  =  N , andis the indicator functional corresponding to  K  with its subdifferential. The lower order multi-valued operatoris 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 allfor 1&lt; p &lt; 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.

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