Extremal Solutions of Multi-Valued Variational Inequalities in Plane Exterior Domains

Let Ω = ℝ2 \ B̅(̅0̅,̅1̅)̅ 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 ∈ K and η ∈ F() such that ⟨− Δ , − ⟩ ≥ ⟨η , − ⟩, ∀ ∈ K, where K is a closed convex subset of the Hilbert space X = D1,20(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 : ℝ → 2ℝ \ {Ø}, and the (single-valued) coefficient α : Ω → ℝ+ is supposed to decay like ||-2-α with α \u3e 0. Unlike in the situation of higher-dimensional exterior domain, that is ℝN with B̅(̅0̅,̅1̅)̅ 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