An ODE reduction method for the semi-Riemannian Yamabe problem on space forms. (August 2021)

• Record Type: Journal Article
• Authors: Fernández, Juan Carlos; Palmas, Oscar
• Is Part Of: Nonlinear analysis. Volume 209(2021)
• Publication Date: 2021-08
• Subjects: primary 34B16 35B06 53C21 53C50 58J45 -- secondary 35B08 35B33 35B44 53C40; Semi-Riemannian Yamabe problem -- Generalized Emden–Fowler equations -- Blowing-up solutions -- Nodal solutions -- Semi-Riemannian isoparametric hypersurfaces; 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.2021.112342
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the semi-Riemannian Yamabe type equations of the form − □ u + λ u = μ | u | p − 1 u on M where M is either the semi-Euclidean space or the pseudosphere of dimension m ≥ 3, □ is the semi-Riemannian Laplacian in M, λ ≥ 0, μ ∈ R ∖︀ { 0 } and p > 1 . Using semi-Riemannian isoparametric functions on M, we reduce the PDE into a generalized Emden–Fowler ODE of the form w ′ ′ + q ( r ) w ′ + λ w = μ | w | p − 1 w on I, where I ⊂ R is [ 0, ∞ ) or [ 0, π ], q ( r ) blows-up at 0 and w is subject to the natural initial conditions w ′ ( 0 ) = 0 in the first case and w ′ ( 0 ) = w ′ ( π ) = 0 in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.