LAUSR

Abstract Let Ω = R 2 ∖ B ( 0 , 1 ) ‾ be the exterior of the closed unit disc in the plane. In this paper we prove existence and enclosure results of multi-valued variational inequalities in Ω of the form: Find u ∈ K and η ∈ F ( u ) such that 〈 − Δ u , v − u 〉 ≥ 〈 a η , v − u 〉 , ∀ v ∈ K , where K is a closed convex subset of the Hilbert space X = D 1 , 2 0 ( Ω ) which is the completion of C c ∞ ( Ω ) with respect to the ‖ ∇ ⋅ ‖ 2 , Ω -norm. The lower order multi-valued operator F is generated by an upper semicontinuous multi-valued function f : R → 2 R ∖ { ∅ } , and the (single-valued) coefficient a : Ω → R + is supposed to decay like | x | − 2 − α with α > 0 . Unlike in the situation of higher-dimensional exterior domain, that is R N ∖ B ( 0 , 1 ) ‾ with N ≥ 3 , the borderline case N = 2 considered here requires new tools for its treatment and results in a qualitatively different behaviour of its solutions. We establish a sub-supersolution principle for the above multi-valued variational inequality and prove the existence of extremal solutions. Moreover, we are going to show that classes of generalized variational-hemivariational inequalities turn out to be merely special cases of the above multi-valued variational inequality.