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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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