Abstract : This study concerns the existence of positive solution for the following nonlinear system $$ \left\{\begin{array}{ll} -div(|x|^{-ap}\,|\nabla u|^{p-2}\,\nabla u) = |x|^{-(a+1)p+c_{1}}\,(\alpha_{1} f(v)+\beta_{1} h(u)), x\in \Omega,\\ -div(|x|^{-bq}\,|\nabla v|^{q-2}\,\nabla v) = |x|^{-(b+1)q+c_{2}}\,(\alpha_{2} g(u)+\beta_{2} k(v)), x\in \Omega,\\ u = v = 0, x\in\partial \Omega, \end{array}\right. $$ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ with $0\in \Omega,$ $1
