In this paper, we prove existence results for quasilinear parabolic bilateral variational inequalities of the form: Find u ∈ K ⊂ X with u(・, 0) = 0 satisfying 0 ∈ ut − Δpu + aF(u) + ∂IK(u) in X∗ in the unbounded cylindrical domain Q = RN × (0, τ ), where Δp is the p-Laplacian acting on X = Lp(0, τ ;D1,p(RN)) with its dual space X∗, and with D1,p(RN) denoting the Beppo-Levi space (or homogeneous Sobolev space). The bilateral constraint is represented by the closed convex set K ⊂ X given by K = {v ∈ X : ϕ(x, t) ≤ v(x, t) ≤ ψ(x, t) for a.a. (x, t) ∈ Q} and IK is the indicator function related to K with ∂IK denoting its subdifferential in the sense of convex analysis. The main goal and the novelty of this paper is to prove existence and directedness results without assuming coercivity conditions on the operator −Δp + aF : X → X∗, and without supposing the existence of sub- and supersolutions. Moreover, additional difficulties we are faced with arise due to the lack of compact embedding of D1,p(RN) into Lebesgue spaces Lσ(RN), and the fact that the domain K of ∂IK has empty interior, which prevents us to use recent results on evolutionary variational inequality. Instead our approach is based on an appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as pseudomontone operator theory.
