Radial non-positive solutions for nonlinear equation \(\Delta u+| x| ^{\ell}| u| ^{p-1}u=0\) on the ball
type: Item
instance of: scholarly article
title: Radial non-positive solutions for nonlinear equation \(\Delta u+| x| ^{\ell}| u| ^{p-1}u=0\) on the ball
zbMATH Open document ID: 0653.35029
author: Ken'ichi Nagasaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: The author studies the radial solutions of the nonlinear boundary value problem \[ (P)\quad \Delta u+| x|^{\ell}| u|^{p- 1}u=0\quad in\quad \Omega =\{x| \quad | x| <1\}\subset {\mathbb{R}}\quad n;\quad u=0\quad on\quad \partial \Omega, \] where \(n\geq 3\), \(\ell \geq 0\) and \(p>1\). The result proved is the following. If \(p\in (1,(n+2+2\ell)/(n-2)),\) for each positive integer k there exists a unique radial solution \(u=u(r)\) of (P), such that \(u(0)>0\) and u(r) has exactly k-1 zeros in(0,1). If \(p\in [(n+2+2\ell)/(n-2),+\infty),\) there exists no radial solution of (P) except the trivial one.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4065507
zbMATH Keywords: non-existence
zbMATH Keywords: existence
zbMATH Keywords: radial solutions
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: unique
MaRDI profile type: Publication
full work available at URL: pjaa.64.5
OpenAlex ID: W2078493874
cites work: Q3244064
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DOI: 10.3792/PJAA.64.5
description: scientific article; zbMATH DE number 4065507