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In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $$\left\}\begin{array}{cccc}-{\Delta }_{p}u=f(x,u,v),\hfill & \text{in}\Omega ,-{\Delta }_{p}v=g(x,u,v),\hfill & \text{in}\Omega ,u=v=0,& \text{on}\partial \Omega ,\end{array}\right.$$ where $-{\Delta }_p$ is the $p$-Laplace operator, $p>1$ and $\Omega$ is a $C^{1,\alpha }$-domain in ${ℝ}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)={\lambda }_1{v}^{p-1}$, $g(x,u,v)={\lambda }_2{u}^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.