An existence theorem for positive solutions of semilinear elliptic equations
type: Item
instance of: scholarly article
title: An existence theorem for positive solutions of semilinear elliptic equations
zbMATH Open document ID: 0624.35034
DOI: 10.1007/BF00251358
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let \(f\in C^ 1\), \(f(p_ 0)=0\) for some \(p_ 0>0\), and \(F(u)=\int^{u}_{0}f(s)ds\). The authors prove the equivalence of the following three statements: (1) There exist positive solutions of \(\Delta u+f(u)=0\) in \(B_ R\), the n-ball of radius R, and \(u=0\) on \(\partial B_ R\) for which u(0) is arbitrarily close to \(p_ 0\), \(u(0)<p_ 0\). (2) There exist positive solutions of \(u''+f(u)=0\) in \(B_ 1\), \(u=0\) on \(\partial B_ 1\) for which u(0) is arbitrarily close to \(p_ 0\), \(u(0)<p_ 0\). (3) \(F(p)<F(p_ 0)\) for \(0\leq p<p_ 0\). The argument is based on the radial symmetry deduced by \textit{B. Gidas, W. M. Ni}, and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)] and is related to results of \textit{P. Hess} [Commun. Partial Differ. Equations 6, 951-961 (1981; Zbl 0468.35073)].
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4013144
zbMATH Keywords: positive solutions
zbMATH Keywords: radial symmetry
author: Joel Smoller
author: Arthur G. Wasserman
reviewed by: Philip Schaefer
MaRDI profile type: Publication
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: Symmetry and related properties via the maximum principle
cites work: On the Existence of Positive Solutions of Semilinear Elliptic Equations
cites work: Global bifurcation of steady-state solutions
cites work: Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
full work available at URL: https://doi.org/10.1007/bf00251358
OpenAlex ID: W1994640183
description: scientific article; zbMATH DE number 4013144