A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains. (September 2017)

• Record Type: Journal Article
• Authors: Vélez-Santiago, Alejandro
• Is Part Of: Nonlinear analysis. Volume 160(2017)
• Page Start: 191
• Page End: 210
• Publication Date: 2017-09
• Subjects: 35J62 -- 35J92 -- 35D30 -- 35B45 -- 35B05 -- 35B65; Neumann boundary conditions -- W1, p-extension domain -- Upper d-set -- Weak solutions -- A priori estimates -- Hölder continuity -- Sub-supersolution method -- Leray–Schauder degree theory; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.05.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let Ω ⊆ R N 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 Δ p, 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.