This study concerns the existence of positive solution for the following nonlinear system $${-div(|x|^{-ap}|{ abla}u|^{p-2}{ abla}u)=|x|^{-(a+1)p+c_1}({alpha}_1f(v)+{eta}_1h(u)),x{in}{Omega},\-div(|x|^{-bq}|{ abla}v|q^{-2}{ abla}v)=|x|^{-(b+1)q+c_2}({alpha}_2g(u)+{eta}_2k(v)),x{in}{Omega},\u=v=0,x{in}{partial}{Omega}$$, where ${Omega}$ is a bounded smooth domain of $mathbb{R}^N$ with $0{in}{Omega}$, 1 < $p,q$ < N, $0{{leq}}a