Uniqueness results for higher order Lane-Emden systems. (September 2020)

• Record Type: Journal Article
• Authors: Cassani, Daniele; Schiera, Delia
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35J48 -- 35A02 -- 35B06; Higher order elliptic operators -- Variational and non-variational elliptic systems -- Gidas–Ni–Nirenberg type results; 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.2020.111871
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet's: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier's boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.