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A singular Gierer-Meinhardt system of elliptic equations. (English) Zbl 0969.35062
The singular elliptic system $$-\Delta u= -u+{u\over v},\quad -\Delta v= -\alpha v+{u\over v}\tag+$$ is studied in a bounded smooth domain $\Omega\subset \bbfR^n$ under homogeneous Dirichlet boundary conditions $u|_{\partial\Omega}= v|_{\partial\Omega}= 0$. Here $\alpha>0$ is a constant. The system $(+)$ is a special case of the so-called “Gierer-Meinhardt”-system from mathematical biology (morphogenesis, predator-prey-interactions, etc.), which is usually studied under Neumann conditions, see e.g. the review article [{\it W.-M. Ni}, Notices Am. Math. Soc. 45, No. 1, 9-18 (1998; Zbl 0917.35047)]. In the latter case, in the framework of positive solutions the singularity in $(+)$ doesn’t become apparent, which is in sharp contrast with the present paper. The authors prove existence of positive solutions $u,v\in C^1(\overline\Omega)\cap C^2(\Omega)$ with help of Schauder’s fixed point theorem. Refined invariant subsets of $C^1(\overline\Omega)\times C^1(\overline\Omega)$ have to be constructed, where the cases $\alpha<1$ and $\alpha>1$ have to be destinguished.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35J45Systems of elliptic equations, general (MSC2000)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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References:
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