Existence of positive solutions for a nonvariational quasilinear elliptic system
type: Item
instance of: scholarly article
title: Existence of positive solutions for a nonvariational quasilinear elliptic system
zbMATH Open document ID: 0964.35049
published in: Journal of Differential Equations
publication date: 2001-07-02T00:00:00Z
review text: Let \(\Omega\) be an open ball with center \(0\) in \({\mathbb R}^N\) and \(\Delta_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)\) be the \(p\)-Laplacian of \(u\). The authors establish the existence of a positive radial solution of the boundary value problem \[ -\Delta u= u^\alpha u^\beta\quad\text{on }\Omega,\qquad -\Delta v= u^\gamma v^\delta\quad\text{on }\Omega,\qquad u,v\equiv 0\quad\text{on }\partial\Omega \] under, e.g., the following hypotheses: \(1<p\), \(q<N\), \(\alpha, \beta, \gamma, \delta\geq 0\), \(\beta\gamma>D:=(p-1-\alpha)(q-1-\delta)\), \(\min\{p-1-\alpha,q-1-\delta\}>0\), \(\max\{\sigma_1-{{N-p}\over{p-1}},\sigma_2-{{N-q}\over{q-1}}\}\geq 0\). Here \(\sigma_1:=[\beta q-p(q-1-\delta)]/D\) and \(\sigma_2:=[\gamma p -q(p-1-\alpha)]/D\). These results generalize earlier work by \textit{Ph. Clément, R. Manásevich} and \textit{E. Mitidieri} [Commun. Partial Differ. Equations 18, No. 12, 2071-2106 (1993; Zbl 0802.35044)]. A priori bounds are obtained by a blow-up method already used in this paper, and a degree argument is then applied. Also, some nonexistence results for the ground state are derived in case that the above system is considered on \({\mathbb R}^N\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1530280
zbMATH Keywords: systems involving \(p\)-Laplacian without variational structure
zbMATH Keywords: a priori bounds
zbMATH Keywords: blow-up method
zbMATH Keywords: degree argument
zbMATH Keywords: nonexistence
author: Philippe Clément
author: Jacqueline Fleckinger
author: Enzo Mitidieri
author: François de Thélin
reviewed by: Georg Hetzer
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3805
OpenAlex ID: W2081563733
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DOI: 10.1006/JDEQ.2000.3805
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description: scientific article; zbMATH DE number 1530280