Existence and nonexistence of radial positive solutions of superlinear elliptic systems
type: Item
instance of: scholarly article
title: Existence and nonexistence of radial positive solutions of superlinear elliptic systems
zbMATH Open document ID: 1034.35040
DOI: 10.5565/PUBLMAT_45201_06
author: Abdelaziz Ahammou
published in: Publicacions Matemàtiques
publication date: 2001-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41435
review text: The author studies the existence of radial solutions of \[ \begin{cases} -\Delta_p u=a(x)\left\vert u\right\vert^{\alpha-1}u +b(x)\left\vert v\right\vert^{\beta-1}v &\text{on } \Omega,\\ -\Delta_q v=c(x)\left\vert u\right\vert^{\gamma-1}u +d(x)\left\vert v\right\vert^{\delta-1}v &\text{on } \Omega,\end{cases}\tag{1} \] subject to homogeneous Dirichlet conditions. \(\Omega\) is either \({\mathbb R}^N\) or an open ball in \({\mathbb R}^N\) centered at 0, \(\Delta_r\) denotes the \(r\)-Laplacian, and \(a\), \(b\), \(c\), \(d\) are continuous function with positive infima. The author establishes two existence results (positive, radial solution), if \(p-1<\alpha\), \(q-1<\delta\) and \(\Omega\) is bounded, and a nonexistence result for \(\Omega={\mathbb R}^N\) (Liouville Theorem). The latter is obtained by means of a blow-up argument due to \textit{B. Gidas} and \textit{J. Spruck} [Commun. Partial Differ. Equ. 6, 883--901 (1981; Zbl 0462.35041)]. The existence results rely on a priori estimates.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1740174
zbMATH Keywords: p-Laplacian
zbMATH Keywords: superlinear
zbMATH Keywords: radial solutions
zbMATH Keywords: blow-up arguments
zbMATH Keywords: topological degree
reviewed by: Georg Hetzer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1740174