Moser iteration applied to elliptic equations with critical growth on the boundary. (March 2019)

• Record Type: Journal Article
• Authors: Marino, Greta; Winkert, Patrick
• Is Part Of: Nonlinear analysis. Volume 180(2019)
• Page Start: 154
• Page End: 169
• Publication Date: 2019-03
• Subjects: 35J60 -- 35B45 -- 35J25; Moser iteration -- Boundedness of solutions -- A priori bounds -- Elliptic operators of divergence type -- Critical growth on the boundary; 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.2018.10.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper deals with boundedness results for weak solutions of the equation (P) − div A ( x, u, ∇ u ) = B ( x, u, ∇ u ) in Ω, A ( x, u, ∇ u ) ⋅ ν = C ( x, u ) on ∂ Ω, where the functions A : Ω × R × R N → R N, B : Ω × R × R N → R, and C : ∂ Ω × R → R are Carathéodory functions satisfying certain p -structure conditions that have critical growth even on the boundary. Based on a modified version of the Moser iteration we are able to prove that every weak solution of(P) is bounded up to the boundary. Under some additional assumptions on the functions A and C this leads directly to regularity for weak solutions of(P) .