Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds
type: Item
instance of: scholarly article
title: Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds
zbMATH Open document ID: 1180.35232
author: Guangyue Huang
author: Wenyi Chen
published in: Osaka Journal of Mathematics
publication date: 2008-11-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1221656643
review text: Let \((M^n,g_0)\) be the compact Einstein manifold with positive scalar curvature \(R_0\) and \(n\geq 3\). In the present study the authors consider the following nonlinear elliptic equation \[ \Delta_0u-\lambda n+f(u)u^{\frac{n+2}{n-2}}=0,\text{ on }M^n\quad u>0,\text{ on } M^n\tag{1} \] where \(\Delta_0\) is the Laplace-Beltrami operator on \(M^n\) related to \(g_0\). The authors prove that for \(0<\lambda<(n-2) R_0/4(n-1)\) and \(f(n)\leq 0\), and at least one of two inequalities is strict, the only positive solution of (1) is constant.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5363222
zbMATH Keywords: compact Einstein manifolds
zbMATH Keywords: positive solution
zbMATH Keywords: curvature
zbMATH Keywords: semi-linear elliptic equation
MaRDI profile type: Publication
cites work: Elliptic equations with critical exponent on \(S^3\): new non-minimising solutions
cites work: Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations
cites work: Q3409165
cites work: Elliptic equations with critical exponent on spherical caps of \(S^{3}\)
cites work: On the Brezis--Nirenberg problem on \(S^3\), and a conjecture of Bandle--Benguria
cites work: Hypersurfaces with constant scalar curvature
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Q4225585
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
cites work: The conjectures on conformal transformations of Riemannian manifolds
cites work: Q3480816
cites work: Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in \(\mathbb R^3\)
description: scientific article; zbMATH DE number 5363222