On variational inequalities on exterior domains with multivalued convection terms

Abstract

In this paper, we study variational inequalities of the form 〈−ΔNu,v−u〉+〈F(u),v−u〉≥0,∀v∈Ku∈K,where ΔNu=div|∇u|N−2∇u is the N-Laplacian, K is a closed convex set in the Beppo–Levi space X=D01,N(Ω), where Ω is the exterior of the closed unit ball B(0,1)¯ in RN (N≥2).We are interested here in the case where F is given by a multivalued function f=f(x,u,∇u) that depends also on the gradient ∇u of the unknown function. We consider a functional analytic framework of the above problem, including conditions on the lower order term f such that the problem can be appropriately formulated in X and a related weighted Orlicz space, and the involved mappings have certain useful monotonicity-continuity properties.Existence of solutions in coercive and noncoercive cases are studied. In the noncoercive case, we follow a sub-supersolution approach and prove the existence of solutions of the above inequality under a multivalued local growth condition of Bernstein–Nagumo type.