The Gierer-Meinhardt system in the entire space with non-local proliferation rates

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows:$\{\begin{array}{cc}-\Delta u+\lambda u=\frac{J⁎u^p}{v^q}+\rho \left(x\right)& \text{ in }{R}^N,N\ge 1,\\ -\Delta v+\mu v=\frac{J⁎u^m}{v^s}& \text{ in }{R}^N.\end{array}$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modeling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J⁎u^p,J⁎u^m$ where the ⁎-symbol denotes the convolution operation in $R^N$ with a kernel $J\in C^1(R^N\setminus \{0\left\}\right)$. In the system, we assume that $0<\rho \in C^{0,\gamma }\left(R^N\right)$ with $\gamma \in (0,1)$, while the parameters satisfy $\lambda ,\mu ,q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel J, we establish the existence and non-existence of classical positive solutions in the function space $C_{loc}^{2,\delta }\left(R^N\right)$. These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.