Radially symmetric solutions of a nonlinear elliptic equation
type: Item
instance of: scholarly article
title: Radially symmetric solutions of a nonlinear elliptic equation
zbMATH Open document ID: 1223.34042
DOI: 10.1155/2011/608576
author: Edward P. Krisner
author: W. C. Troy
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2011-07-22T00:00:00Z
review text: Summary: We investigate the existence and asymptotic behavior of positive solutions of \[ w'' + ((N - 1)/r)w' - |w|^{p-1}w = 0\quad r > 0. \] We focus on the parameter regime \(N > 2\) and \(1 < p < N/(N - 2)\) where the equation has the closed form, positive singular solution \(w_1 = (4 - 2(N - 2)(p - 1)/(p - 1)^2)^{1/(p - 1)}r^{-2/(p - 1)}\), \(r > 0\). Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval \((r_{\min}, r_{\max})\). Our aim is to transform the nonautonomous \(w\) equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval \((0, \infty)\), and which satisfy \(0 < w(r) < w_1(r)\) for all \(r > 0\). An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of ``super singular'' solutions which lie entirely above \(w_1(r)\).
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 5930213
zbMATH Keywords: radially symmetric solutions
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: existence and asymptotic behavior
zbMATH Keywords: positive solutions
Wikidata QID: Q58687877
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/608576
OpenAlex ID: W1980671034
cites work: A very singular solution of the heat equation with absorption
cites work: Singular Solutions of the Heat Equation with Absorption
cites work: Classification of singular solutions of a nonlinear heat equation
cites work: Q3916903
cites work: Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state
cites work: Global existence and blow-up of solutions for a semilinear parabolic system
cites work: Positive and oscillatory radial solutions of semilinear elliptic equations
cites work: Spatial patterns. Higher order models in physics and mechanics
description: scientific article; zbMATH DE number 5930213