Multiplicity of positive radial solutions for an elliptic system on an annulus
type: Item
instance of: scholarly article
title: Multiplicity of positive radial solutions for an elliptic system on an annulus
zbMATH Open document ID: 0959.35051
DOI: 10.1016/S0362-546X(99)00125-X
author: D. R. Dunninger
author: Haiyan Wang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-04-18T00:00:00Z
review text: The authors study the existence of positive radial solutions of the elliptic system \[ \Delta u + \lambda k_1(|x |) f(u,v)=0, \quad \Delta v + \lambda k_2(|x |) g(u,v)=0 \text{ in } \Omega, \] \[ \alpha_1 u + \beta_1 \frac{\partial u}{\partial n}=0, \quad \alpha_2 v + \beta_2 \frac{\partial v}{\partial n}=0 \text{ on } |x |=R_1, \] \[ \gamma_1 u + \delta_1 \frac{\partial u}{\partial n}=0, \quad \gamma_2 v + \delta_2 \frac{\partial v}{\partial n}=0 \text{ on } |x |=R_2, \] where \(\alpha_i\), \(\beta_i\), \(\gamma_i\), \(\delta_i \geq 0\) and \(\gamma_i \beta_i + \alpha_i \gamma_i + \alpha_i \delta_i >0\) for \(i=1, 2.\) Under suitable conditions on the functions that define the system, they prove that there exist \(\lambda^* >0\) for which this problem has at least two positive solutions for \(0< \lambda < \lambda^*\), at least one positive solution for \(\lambda=\lambda^*\) and no positive solution for \(\lambda > \lambda^*\). In the proof degree theory, Green's functions properties and existence results via lower and upper solutions are used.
reviewed by: Alberto Cabada
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1513296
zbMATH Keywords: elliptic system
zbMATH Keywords: positive radial solution
zbMATH Keywords: multiplicity
zbMATH Keywords: upper and lower solutions
MaRDI profile type: Publication
cites work: Positive solutions of nonlinear elliptic systems
cites work: Q4871247
cites work: Existence and multiplicity of positive solutions for elliptic systems
cites work: Q3994493
cites work: Existence of multiple positive solutions of singular boundary value problems
cites work: Existence of positive radial solutions for elliptic systems
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description: scientific article; zbMATH DE number 1513296