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A singular Gierer-Meinhardt system of elliptic equations. (English) Zbl 0969.35062

The singular elliptic system

$-{\Delta }u=-u+\frac{u}{v},\phantom{\rule{1.em}{0ex}}-{\Delta }v=-\alpha v+\frac{u}{v}\phantom{\rule{2.em}{0ex}}\left(+\right)$

is studied in a bounded smooth domain ${\Omega }\subset {ℝ}^{n}$ under homogeneous Dirichlet boundary conditions ${u|}_{\partial {\Omega }}{=v|}_{\partial {\Omega }}=0$. Here $\alpha >0$ is a constant. The system $\left(+\right)$ 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 $\left(+\right)$ doesn’t become apparent, which is in sharp contrast with the present paper.

The authors prove existence of positive solutions $u,v\in {C}^{1}\left(\overline{{\Omega }}\right)\cap {C}^{2}\left({\Omega }\right)$ with help of Schauder’s fixed point theorem. Refined invariant subsets of ${C}^{1}\left(\overline{{\Omega }}\right)×{C}^{1}\left(\overline{{\Omega }}\right)$ 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 for nonlinear operators on topological linear spaces