Existence results for highly discontinuous implicit elliptic equations

Abstract

Let n ∈ N, with n ≧ 3, let p ∈]n/2,+∞[, and let Ω ⊆ Rⁿ be a bounded domain with smooth boundary. Let Y ⊆ Rⁿ, and let φ : Ω x R^h → R and Ψ : Y → R be two given functions, with Ψ continuous. We study the existence of strong solutions u = (u₁, ..., u_h) ∈ W^2,p (Ω,R^h) ∩ W₀^1,p (Ω,R^h) of the implicit elliptic equation Ψ(-Δu) = φ(x, u), where Δu = (Δu₁, Δu₂, ..., Δu_h). We prove existence results where φ is allowed to be highly discontinuous in both variables. In particular, a function φ(x,z) satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points z ∈ R^h.