Positive clustered layered solutions for the Gierer-Meinhardt system
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.
scientific article; zbMATH DE number 5306520