Singular solutions to k-Hessian equations with fast-growing nonlinearities. (September 2022)

• Record Type: Journal Article
• Authors: do Ó, João Marcos; Shamarova, Evelina; da Silva, Esteban
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: 34B15 -- 34C23 -- 35A24 -- 35J15 -- 35J62 -- 35J92; k-Hessian equation -- Singular solution -- Multiplicity of regular solutions -- Number of intersection points -- Bifurcation diagrams -- Liouville–Bratu–Gelfand problem; 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.2022.113000
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study a class of elliptic problems, involving a k -Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular and analyze the intersection number depending on the parameter k and the dimension d .