On a k-Hessian equation with a weakly superlinear nonlinearity and singular weights. (January 2020)

• Record Type: Journal Article
• Authors: Feng, Meiqiang; Zhang, Xuemei
• Is Part Of: Nonlinear analysis. Volume 190(2020)
• Publication Date: 2020-01
• Subjects: 35J60 -- 35J96; k-Hessian equation -- Boundary blow up -- Sub-supersolution method -- k-convex solution -- Weakly superlinear nonlinearity; 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.2019.111601
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Consider the existence, nonexistence and global estimates of k -convex solutions to the boundary blow-up k -Hessian problem S k ( D 2 u ( x ) ) = H ( x ) [ u ( x ) ] k [ ln u ( x ) ] β > 0 for x ∈ Ω, u ( x ) → + ∞ as dist ( x, ∂ Ω ) → 0 . Here k ∈ { 1, 2, …, N }, S k ( D 2 u ) is the k -Hessian operator, β > 0, Ω is a smooth, bounded, strictly convex domain in R N ( N ≥ 2 ), and H ( x ) is a positive weight function which is singular near the boundary ∂ Ω . We first give the existence and nonexistence results of k -convex solution to the above boundary blow-up problem on a larger range of H and β . Then we show that there is a k -convex solution provided that H ( x ) grows fast near ∂ Ω and u k [ ln u ] β grows slow at ∞ . It turns out that this case is more difficult to handle than the case in which H ( x ) grows slow near ∂ Ω and u k [ ln u ] β grows fast at ∞ . This needs some new ingredients in the arguments.