A quasi-linear Neumann problem of Ambrosetti-Prodi type on extension domains

Let Ω⊆RN be a bounded domain with the extension property, whose boundary Γ≔∂Ω is an upper d-set (with respect to a measure μ), for N≥2 and d∈(N−p,N). We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator Δp, with Neumann boundary conditions. Using a priori estimates, regularity theory, a sub-supersolution method, and the Leray-Schauder degree theory, we obtain a necessary condition for the non-existence of solutions (in the weak sense), the existence of at least one minimal solution, and the existence of at least two distinct solutions. Moreover, we establish global Hölder continuity for weak solutions of the Neumann problem of Ambrosetti-Prodi type on a large class of “bad” domain, extending the solvability of this type of elliptic problem for the first time, to a wide class of non-smooth domains.