NONCOERCIVE ELLIPTIC BILATERAL VARIATIONAL INEQUALITIES IN THE HOMOGENEOUS SOBOLEV SPACE D1,p(ℝN).

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δ_pu+aF(u) + ...I_K(u) ∈ V* in the whole ℝ ^N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ ^N}, where Δ_p is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D^1,p(ℝ^N) with 1 < p < N, and I_K : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...I_K. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝ^N x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- ^(N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δ_p + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into L^q(ℝ^N) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝ^N. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]