A class of singular Gierer--Meinhardt systems of elliptic boundary value problems (Q1888601)

The author deals with the following steady-state problems \[ \begin{cases} \Delta u-\alpha u+\rho{u^p\over v^q}+\rho= 0\quad &\text{in }\Omega,\;u|_{\partial\Omega}= 0,\\ \Delta v-\beta v+\rho{u^p\over v^q}= 0\quad &\text{in }\Omega,\;v|_{\partial\Omega}= 0,\end{cases}\tag{1} \] where \(\alpha\), \(\beta\) are positive parameters and \(\rho= \rho(x)\) is the source distributions. The goal of the author is to establish a mathematical theory for singular elliptic systems (1). To this end, the author uses the Schauder fixed point theory and upper-lower solution methods.