Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition

This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions:

$$\left\{\begin{tabular}{ll} $\displaystyle\Delta u- \alpha u +\fra... ...yle u=0,\; v=0$ \quad & $\mbox{\rm on } \partial\Omega,$ \end{tabular} \right.$$

where $\Omega\subset\mathbb{R}^N$ ($N\geq 1$) is a smooth bounded domain, $\rho(x)\geq 0$ in $\Omega$ and $\alpha,\beta\geq 0$. We are mainly interested in the case of different source terms, that is, $(p,q)\neq (r,s)$. Under appropriate conditions on the exponents $p,q,r$ and $s$ we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.