Nonexistence of subsolutions of a nonlinear elliptic equation on bounded domains in a Riemannian manifold
type: Item
instance of: scholarly article
title: Nonexistence of subsolutions of a nonlinear elliptic equation on bounded domains in a Riemannian manifold
zbMATH Open document ID: 0917.35032
author: Shin Kato
published in: Hiroshima Mathematical Journal
publication date: 1999-07-25T00:00:00Z
review text: Given a Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) and a smooth function \(f\) on \(M\), \(f\) is the scalar curvature of a metric \(\widetilde g\) conformal to \(g\) if and only if there is a smooth positive solution \(u\) of \[ - {{4(n-1)}\over{(n-2)}} \Delta_g u + S_g u = fu^{(n+2)/(n-2)} \quad\text{on}\quad M, \tag{\(*\)} \] where \(\Delta_g\) and \(S_g\) are the Laplace-Beltrami operator and scalar curvature of \((M,g)\) respectively. A large amount of research has been devoted to understanding the solvability of \((*)\) under various assumptions on \((M,g)\) and \(f\). Here the author considers a class of equations including \((*)\) on certain types of open Riemannian manifolds and proves some nonexistence results in the case that \(f\) is nonpositive and satisfies a suitable growth assumption near \(\partial M\). Of particular interest is the fact that there is no assumption on the dimension or regularity of \(\partial M\). The main theorems generalize and include a number of earlier results as special cases.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1251457
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: scalar curvature equation
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206126675
OpenAlex ID: W1734630047
Wikidata QID: Q115222449
DOI: 10.32917/HMJ/1206126675
description: scientific article; zbMATH DE number 1251457