Boundary behavior of k-convex solutions for singular k-Hessian equations. (November 2018)

• Record Type: Journal Article
• Authors: Sun, Huayuan; Feng, Meiqiang
• Is Part Of: Nonlinear analysis. Volume 176(2018)
• Page Start: 141
• Page End: 156
• Publication Date: 2018-11
• Subjects: Existence -- Boundary behavior -- k-convex solution -- Singular k-Hessian equations -- Regular variation 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.2018.06.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We discuss the existence and boundary behavior of k -convex solution to the singular k -Hessian problem S k ( D 2 u ( x ) ) = b ( x ) f ( − u ( x ) ), x ∈ Ω, u ( x ) = 0, x ∈ ∂ Ω, where S k ( D 2 u ) ( k ∈ { 1, 2, …, n } ) is the k -Hessian operator, Ω ⊂ R n ( n ≥ 2 ) is a smooth bounded strictly convex domain. Here the weight function b ( x ) is not necessarily bounded on ∂ Ω . Another interest is that f ( u ) → ∞ as u → 0 . Our approach mainly relies on Karamata's regular variation theory and the construction of suitable sub- and super-solutions.