Singular Gierer-Meinhardt systems of elliptic boundary value problems

We establish existence results for singular Gierer-Meinhardt elliptic systems with zero Dirichlet boundary conditions. Gierer-Meinhardt systems are model problems for pattern formations of spatial tissue structures of morphogenesis. The mathematical difficulties are that the system becomes singular near the boundary and it is non-quasimonotone. We show the existence of positive solutions for the activator-inhibitor model with common sources.