The singular elliptic system
is studied in a bounded smooth domain under homogeneous Dirichlet boundary conditions . Here 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 with help of Schauder’s fixed point theorem. Refined invariant subsets of have to be constructed, where the cases and have to be destinguished.