Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
type: Item
instance of: scholarly article
title: Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
zbMATH Open document ID: 0582.35046
DOI: 10.1007/BF01468138
published in: Communications in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: This article studies positive solutions of the Dirichlet problem \[ (1)\quad \Delta u(x)+f(u(x))=0,\quad x\in D^ n,\quad u(x)=0,\quad x\in \partial D^ n \] where \(D^ n\) is an n-ball. Since positive solutions of (1) are radially symmetric the authors consider a general ODE-problem \[ (2)\quad u''+g(u,u',t)=0,\quad u'(0)=0,\quad u(R)=0. \] For (2) the so called time-map T(p) is defined on the set \(A=\{p\in R^+:\quad u(t,p)=0\) for some \(t>0\}\) via \(T(p)=\min \{t>0:\quad u(t,p)=0\}.\) (u(t,p) is the solution to (2) with \(u(0,p)=p.)\) u is a positive solution of (2) iff \(u=u(\cdot,p)\), \(p\in A\) and \(R=T(p)\). Uniqueness follows from A being connected and \(T'(p)>0\) or \(<0\). For \(g(u,u',t)=(n-1)/t\cdot u'+f(u)\) the authors give \((3)\quad (f(u)/u)'<0\) as a condition for \(T'>0\), and \((4)\quad (f(u)/u)'>0,\quad f''(u)<0\) for \(T'<0.\) For existence, conditions on the growth of f at \(+\infty\) are given for which \(A\neq \emptyset\), like \(f(u)/u^ k\to 1\) for \(u\to \infty\) and \(2/(n-2)>k.\) Proofs use comparison and phase plane arguments. As the authors point out the usual method of finding sub- and supersolutions does not apply if e.g. (4) holds for f, since then all positive solutions are unstable. Nondegeneracy of a solution u (i.e. 0 is not in the spectrum of the linearization of (1) about u) is shown to be equivalent to \(u(x)=u(| x|,p),\quad T'(p)\neq 0,\quad u'(T(p),p)\neq 0.\)
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3931550
zbMATH Keywords: positive solutions
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: time-map
zbMATH Keywords: Uniqueness
zbMATH Keywords: existence
zbMATH Keywords: comparison
zbMATH Keywords: phase plane arguments
zbMATH Keywords: Nondegeneracy
author: Joel Smoller
author: Arthur G. Wasserman
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01468138
OpenAlex ID: W2063792976
description: scientific article; zbMATH DE number 3931550