Ground states of \(-\Delta u=f(u)\) and the Emden-Fowler equation
type: Item
instance of: scholarly article
title: Ground states of \(-\Delta u=f(u)\) and the Emden-Fowler equation
zbMATH Open document ID: 0606.35029
DOI: 10.1007/BF00279955
author: F. V. Atkinson
author: Lambertus A. Peletier
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: Radially symmetric solutions of nonlinear Poisson equations in \({\mathbb{R}}^ n\) (n\(\geq 2):\) \[ (I)\quad -\Delta u=f(u),\quad u>0\quad in\quad {\mathbb{R}}^ n;\quad u(x)\to 0\quad as\quad x\to \infty, \] are considered. The search of radial solutions of Problem I leads to the ODE problem: \[ (II)\quad u''+((n-1)/r)u'+f(u)=0,\quad 0<r<\infty,\quad u(r)>0,\quad 0<r<\infty,\quad u'(0)=0,\quad u(r)\to 0\quad as\quad r\to 0. \] The paper completes the ODE approach to (I) by placing the problem in the context of the rather more general Emden-Fowler equation of the type: \[ (III)\quad y''+a(t)f(y)=0,\quad y>0,\quad -\infty \leq A<t<\infty,\quad y(A)=0,\quad \lim_{t\to \infty}y(t)\quad exists. \] Conditions on a(t) and f(y) are formulated which guarantee the existence of solutions of Problem III. This yields in particular results about Problem II but also about related problems.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3979600
zbMATH Keywords: Radially symmetric solutions
zbMATH Keywords: nonlinear Poisson equations
zbMATH Keywords: Emden-Fowler equation
zbMATH Keywords: existence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3979600