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Zbl 0969.35062
Choi, Y.S.; McKenna, P.J.
A singular Gierer-Meinhardt system of elliptic equations. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17, No.4, 503-522 (2000). ISSN 0294-1449

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.\par 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.
[Hans-Christoph Grunau (Bayreuth)]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35A05 General existence and uniqueness theorems (PDE)
35J45 Systems of elliptic equations, general
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: singular nonlinearity; semilinear elliptic system; positive solutions

Citations: Zbl 0917.35047

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