On nodal solutions of the Yamabe equation on products
type: Item
instance of: scholarly article
title: On nodal solutions of the Yamabe equation on products
zbMATH Open document ID: 1177.53040
author: Jimmy Petean
published in: Journal of Geometry and Physics
publication date: 2009-10-01T00:00:00Z
review text: Usually, one looks for positive solutions of the Yamabe equation. However, solutions which change signs, called \textit{nodal}, are also important. In this case, the equation takes the form \(L_g(u)=\lambda|u|^{p-2}u\), where \(L_g\) is the Yamabe operator associated to the Riemannian metric \(g\). The main result of this paper is that the Yamabe equation associated to a Riemannian product metric has at least one nodal solution. The particular case when one of the factors in the product is the circle is separately treated and it turns out that there is an infinite number of solutions, explicitly determined.
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 5610178
zbMATH Keywords: Yamabe invariant
zbMATH Keywords: second Yamabe invariant
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: Riemannian product
reviewed by: Liviu Ornea
MaRDI profile type: Publication
full work available at URL: j.geomphys.2009.07.003
OpenAlex ID: W2092702137
cites work: Q3275400
cites work: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
cites work: Q5542608
cites work: Conformal deformation of a Riemannian metric to constant scalar curvature
cites work: Q5699014
cites work: Nodal solutions on Riemannian manifolds
cites work: Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth
cites work: MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS
cites work: The second Yamabe invariant
cites work: Surgery and the Yamabe invariant
cites work: Yamabe metrics on cylindrical manifolds
cites work: On Yamabe constants of Riemannian products
cites work: Q3480816
DOI: 10.1016/J.GEOMPHYS.2009.07.003
description: scientific article; zbMATH DE number 5610178