In this paper, the existence results of positive solutions for the semiliner elliptic system \[ \begin{cases} -\text{div} (|\nabla u_i|^{p-2} \nabla u_i) - \mu \frac{|u_i|^{p-2}u_i}{|x|^p} \\ = \frac{1}{p^*} F_{u_i}(u_1,\ldots,u_k) + \frac{|u_i|^{p^*(t)-2}u_i}{|x|^t} + \lambda \frac{|u_i|^{p-2}u_i}{|x|^s}, \quad x \in \Omega, \\ u_i=0 \quad \text{on} \;\partial \Omega, \\ 1 \leq i \leq k, \end{cases} \] are established, where \(0 \in \Omega\) is a bounded domain in \({\mathbb R}^N\) with the smooth boundary \(\partial \Omega\), \(N \geq 3\), \(10\), \(0 \leq t