The scalar curvature equation on \(\mathbb{R}^ n\) and \(S^ n\)
type: Item
instance of: scholarly article
title: The scalar curvature equation on \(\mathbb{R}^ n\) and \(S^ n\)
zbMATH Open document ID: 0865.35044
author: Gabriele Bianchi
published in: Advances in Differential Equations
publication date: 1997-07-10T00:00:00Z
review text: This paper discusses the equation \(\Delta u+\lambda K(x)u^p=0\) in \(\mathbb{R}^n\) when \(n>2\), \(p\) is the critical exponent \((n+2)/(n-2)\), and \(K\in C^2(\mathbb{R}^n)\) is given. Positive solutions to this equation give a metric on \(\mathbb{R}^n\) which is conformal to the standard one and which has scalar curvature \(K\). Some sufficient conditions are given for this equation to have a solution. The complete description of these conditions is rather complicated, so this review presents a summary. In Theorem 1, near a maximum point of \(K\), the gradient of \(K\) should vanish to a sufficiently high order and, at critical points of \(K\) where \(K\) is sufficiently close to its maximum, \(\Delta K\) is nonzero. In Theorem 2, \(K\) should have only finitely many critical points, there is a strict local maximum point \(z\) of \(K\) at which \(\Delta K\) is nonzero, and, at critical points near \(z\), \(\Delta K\) is nonzero. Unlike most previous works, no symmetry is assumed for \(K\) although recently Noussair and Yang have proved some results for the more general equation \(-\Delta u=K(x)u^p+f(x)\) under different hypotheses on \(K\) which do not require any symmetry. The key to Bianchi's approach is to study the problem on the unit \(n\)-sphere \(S^n\) (as well as the corresponding scalar curvature equation \(\Delta u-{n(n-2)\over 4}u+K(u)u^p=0\)). The proofs involve studying the flow lines of the functional \[ {\mathcal F}(u)={1\over\int_{S^n} K|u|^{p+1}dv}, \] where \(dv\) is the volume element on \(S^n\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 896502
zbMATH Keywords: positive solutions
zbMATH Keywords: scalar curvature equation
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 896502