Liouville property and existence of entire solutions of Hessian equations. (October 2022)

• Record Type: Journal Article
• Authors: Wang, Cong; Bao, Jiguang
• Is Part Of: Nonlinear analysis. Volume 223(2022)
• Publication Date: 2022-10
• Subjects: 35J60 -- 53C24; Liouville property -- Entire solutions -- Existence and uniqueness -- Hessian equations -- Prescribed asymptotic behavior; 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.2022.113020
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge–Ampère equations. Our approach also makes the prescribed asymptotic order optimal within the range preset in exterior Dirichlet problems. In addition, we show a Liouville type result for k -convex solutions. This partly removes the ( k + 1 ) - or n -convexity restriction imposed in existing work.