Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

Abstract

This paper is concerned with the stationary Gierer-Meinhardt system with singularity: \begin{document}$\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, & d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0, & x\in\Omega, u(x)>0, v(x)>0, & x\in \Omega, \displaystyle u(x)=v(x)=0, & x\in\partial\Omega, \end{array}\right.$\end{document} where $-\infty