Implicit highly discontinuous boundary value problems involving the p-Laplacian

Abstract

Let n ∈ N, with n ⋝ 2, and let p ∈]n,+∞[. Let Ω ⊆ Rⁿ be a bounded connected open set, with smooth boundary ∂Ω, and let Y ⊆ R be a closed interval. We study the existence of solutions u ∈ W₀^1,p(Ω) of the implicit equation Ψ(-∆_pu) = f(x,u), where f : Ω ✕ R → R and Ψ : Y → R are two given functions. We establish some existence results where f is allowed to be highly discontinuous in both variables. In particular, a function f(x,z) satisfying the assumptions of our results can be discontinuous, with respect to the second variable, even at all points z ∈ R. As regard Ψ, we only require that it is continuous and locally nonconstant.