Entire positive $ k $-convex solutions to $ k $-Hessian type equations and systems

In this paper, we study the existence of entire positive solutions for the $ k $-Hessian type equation \begin{document}$ {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(u), \ \ x\in \mathbb{R}^n $\end{document} and system \begin{document}$ \begin{cases} {\rm S}_k(D^2u+\alpha I) = p(|x|)f^k(v), \ \ x\in \mathbb{R}^n, \\ {\rm S}_k(D^2v+\alpha I) = q(|x|)g^k(u), \ \ x\in \mathbb{R}^n, \end{cases} $\end{document} where $ D^2u $ is the Hessian of $ u $ and $ I $ denotes unit matrix. The arguments are based upon a new monotone iteration scheme.