Remarks on a semilinear elliptic equation on \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: Remarks on a semilinear elliptic equation on \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0662.35038
DOI: 10.1016/0022-0396(88)90017-4
author: Yi Li
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: We consider the following semilinear elliptic equation \[ (1.1)\quad \Delta u-u+Q| u|^{p-1}u=0\quad in\quad {\mathbb{R}}^ n;\quad u\geq 0,\quad u\not\equiv 0\quad in\quad {\mathbb{R}}^ n,\quad and\quad u\to 0\quad at\quad \infty, \] where p is a constant satisfying \(1<p<(n+2)/(n- 2).\) (For \(n=2\), the number \((n+2)/(n-2)\) is considered to be \(\infty.)\) We study the existence of solutions of equation (1.1) with some restrictions on Q. The symmetry properties and nonexistence of solutions are investigated, we find a class of radially symmetric potentials Q for which positive nonradial solutions of (1.1) exist, and a nonexistence theorem is proved for certain radial potentials.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 4082085
zbMATH Keywords: semilinear
zbMATH Keywords: existence
zbMATH Keywords: symmetry
zbMATH Keywords: nonexistence
zbMATH Keywords: radially symmetric potentials
zbMATH Keywords: positive nonradial
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90017-4
OpenAlex ID: W2018833240
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description: scientific article; zbMATH DE number 4082085