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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 $$\Omega\subset \mathbb{R}^n$$ 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 [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:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J45 Systems of elliptic equations, general (MSC2000) 47H10 Fixed-point theorems
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