Positive clustered layered solutions for the Gierer-Meinhardt system
type: Item
instance of: scholarly article
title: Positive clustered layered solutions for the Gierer-Meinhardt system
zbMATH Open document ID: 1155.35022
published in: Journal of Differential Equations
publication date: 2008-07-31T00:00:00Z
review text: This paper deals with the stationary Gierer-Meinhardt system in a ball of \({\mathbb R}^N\): \[ \begin{cases} \varepsilon^2\Delta u-u+{u^p\over v^q}=0 & \text{in }B_R,\\ \Delta v-v+{u^m\over v^s}=0 & \text{in }B_R,\\ u, v>0 & \text{in } B_R,\\ {\partial u\over \partial \nu}={\partial v\over \partial \nu}=0 & \text{on }\partial B_R. \end{cases} \] Here \(B_R\) is a ball of radius \(R\) in \({\mathbb R}^N\), \(N\geq 2\), and \(\varepsilon>0\) is a small parameter. Under appropriate conditions on \(p,q,m,s\), the authors prove that for any integer \(K\geq 1\), the system has at least one radially symmetric solution \((u_{\varepsilon, K},v_{\varepsilon, K})\) which concentrates at \(K\) spheres. This generalizes the results in \textit{W.-M. Ni} and \textit{J. Wei} [J. Differ. Equations 221, No. 1, 158--189 (2006; Zbl 1090.35023)] in which the special case \(K=1\) is considered.
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 92C37
zbMATH DE Number: 5306520
zbMATH Keywords: clustered layer solutions
zbMATH Keywords: Gierer-Meinhardt system
zbMATH Keywords: pattern formation
zbMATH Keywords: singular perturbations
author: Theodore Kolokolnikov
author: Juncheng Wei
MaRDI profile type: Publication
full work available at URL: j.jde.2008.05.008
OpenAlex ID: W2023584391
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DOI: 10.1016/J.JDE.2008.05.008
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description: scientific article; zbMATH DE number 5306520