Abstract: In this paper, we are concerned with the existence of the positive bounded and blow-up radial solutions of the $(k_{1},k_{2})$-Hessian system \begin{equation*} \begin{split} \left\{\begin{array}{l}{\mathcal{G} (K_{1}^{\frac{1}{k_{1}}}) K_{1}^{\frac{1}{k_{1}}}=b_{1}(|x|) g_{1}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}}, \\ {\mathcal{G}(K_{2}^{\frac{1}{k_{2}}}) K_{2}^{\frac{1}{k_{2}}}=b_{2}(|x|) g_{2}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}},\end{array}\right. \end{split} \end{equation*} where $\mathcal{G}$ is a nonlinear operator, $K_{i}=S_{k_{i}}\left(\lambda\left(D^{2} z_{i}\right)\right)+\psi_{i}(|x|)|\nabla z_{i}|^{k_{i}},i=1,2.$ Under the appropriate conditions on $g_{1}$ and $g_{2}$, our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem. Furthermore, our main results also extend the previous existence results for both the single equation and systems.

Key words: $(k_{1},k_{2})$-Hessian system, nonlinear operator, blow-up, monotone iterative method, gradient