Summary: We consider the existence of positive solutions for the quasilinear elliptic system
\[ \begin{cases} -\Delta u_1=\lambda u_2^{\alpha_1}, &x\in\Omega,\\ -\Delta u_2=\lambda u_3^{\alpha_2}, &x\in\Omega,\\ \quad \cdots\\ -\Delta u_n=\lambda u_1^{\alpha_n}, &x\in\Omega,\\ u_i=0, &x\in\partial\Omega, \quad i=1,2,\dots, \end{cases} \]
where \(\lambda\) is a positive parameter, \(\Delta\) is the Laplacian operator, \(\alpha_i>0\) for \(i=1,2,\dots,n\), and \(\Omega\) is a bounded domain in \(\mathbb R^N\) \((N>1)\) with smooth boundary \(\partial\Omega\). By using the method of sub-super solutions we prove the existence of positive solutions for each \(\lambda>0\).