Existence and nonexistence of positive solutions for a class of superlinear semipositone systems
type: Item
instance of: scholarly article
title: Existence and nonexistence of positive solutions for a class of superlinear semipositone systems
zbMATH Open document ID: 1175.35044
author: Maya Chhetri
author: Petr Girg
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-08-21T00:00:00Z
review text: We consider an elliptic system of the form \[ \begin{aligned} -\Delta u=\lambda f(v) &\quad\text{in }\Omega,\\ -\Delta v=\lambda g(u) &\quad\text{in }\Omega,\\ u=0=v &\quad\text{on }\partial\Omega, \end{aligned} \] where \(\lambda>0\) is a parameter, \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\). Here the nonlinearities \(f,g:[0,\infty)\to\mathbb R\) are \(C_{\text{loc}}^{0,\sigma}\), \(0<\sigma<1\), functions that are superlinear at infinity and satisfy \(f(0)<0\) and \(g(0)<0\). We prove that the system has a positive solution for \(\lambda\) small when \(\Omega\) is convex with \(C^3\) boundary and no positive solution for \(\lambda\) large when \(\Omega\) is a general bounded domain with \(C^{2,\beta}\) boundary. Moreover, we show that there exists a closed connected subset of positive solutions bifurcating from infinity at \(\lambda=0\).
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 5597893
zbMATH Keywords: Laplacian
zbMATH Keywords: semipositone systems
zbMATH Keywords: superlinear
zbMATH Keywords: positive solutions
zbMATH Keywords: nonexistence
zbMATH Keywords: bifurcation from infinity
MaRDI profile type: Publication
full work available at URL: j.na.2009.03.070
OpenAlex ID: W2056703734
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DOI: 10.1016/J.NA.2009.03.070
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description: scientific article; zbMATH DE number 5597893