Positive Solutions for a Singular Gierer–Meinhardt Elliptic System with Sublinear Nonlinearities and Neumann Boundary Conditions

In this paper, a singular stationary Gierer–Meinhardt-type activator–inhibitor system with homogeneous Neumann boundary conditions is investigated. The model incorporates singular interaction terms together with sublinear source nonlinearities, which describe production mechanisms that grow slowly at high concentrations. The main objective is to establish the existence of positive solutions for the associated elliptic system defined on a bounded domain with a smooth boundary. To overcome the lack of a variational structure caused by the singular nonlinearities, an approximation scheme combined with the Galerkin method and a variant of Brouwer’s fixed point theorem is employed. A suitable perturbed problem was introduced in order to regularize the singular terms and obtain uniform a priori estimates. It was proved that, for sufficiently small parameters λ and μ, the system admits a positive weak solution in H1(Ω). Furthermore, lower bounds for the components of the solution were derived, and regularity results were obtained through bootstrap arguments. These results extend existing studies on singular Gierer–Meinhardt systems by incorporating sublinear nonlinearities and Neumann boundary conditions, which are relevant in models of biological pattern formation and diffusion–reaction processes.