Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let Ω be a bounded domain in ℝn (n ≥ 3) having a smooth boundary, let ψ be an essentially bounded real-valued function defined on Ω × ℝh, and let ψ be a continuous real-valued function defined on a given subset Y of ℝh. In this paper, the existence of strong solutions u ∈ W2,p (Ω,ℝh) ∩ W01,p (n/2 < p < +∞) to the implicit elliptic equation ψ(-Δu) = ψ?(x, u), with u = (u1, u2,..., uh) and Δu = (Δu1, Δu2,..., Δuh), is established. The abstract framework where the problem is placed is that of set-valued analysis. © 1996 Kluwer Academic Publishers.