We study the prototype model of the boundary value problem
$$ \begin{array}{rl} {\rm div}(|\nabla u|^{m-2}\nabla u) + u^av^b = 0& \mbox{in }\quad \Omega, \\ {\rm div}(|\nabla v|^{m-2}\nabla v) + u^cv^d = 0& \mbox{in }\quad \Omega, \\ u = v = 0&\mbox{on } \quad\partial\Omega, \end{array} $$
where $\Omega\subset\bm{R}^n$ ( $n\ge2$ ) is a connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge0$ are non-negative numbers. Under appropriate conditions on the exponents $m$ , $a$ , $b$ , $c$ and $d$ , and on the domain $\Omega$ , a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.