Classification and existence of positive entire k-convex radial solutions for generalized nonlinear k-Hessian system

In this paper, we consider the following generalized nonlinear k-Hessian system {G(Sk1k(λ(D2z1)))Sk1k(λ(D2z1))=φ(|x|,z1,z2),x∈ℝN,G(Sk1k(λ(D2z2)))Sk1k(λ(D2z2))=φ(|x|,z1,z2),x∈ℝN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{{\cal G}\left( {S_k^{{1 \over k}}(\lambda ({D^2}{z_1}))} \right)S_k^{{1 \over k}}(\lambda ({D^2}{z_1})) = \varphi (\left| x \right|,{z_1},{z_2}),\,\,\,x \in {\mathbb{R}^N},} \cr {{\cal G}\left( {S_k^{{1 \over k}}(\lambda ({D^2}{z_2}))} \right)S_k^{{1 \over k}}(\lambda ({D^2}{z_2})) = \varphi (\left| x \right|,{z_1},{z_2}),\,\,\,x \in {\mathbb{R}^N},} \cr } \,} \right.$$\end{document} where G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal G}$$\end{document} is a nonlinear operator and Sk (λ(D2z)) stands for the k-Hessian operator. We first are interested in the classification of positive entire k-convex radial solutions for the k-Hessian system if φ(∣x∣, z1, z2) = b(∣x∣)φ(z1, z2) and ψ(∣x∣, z1, z2) = h(∣x∣)ψ(z1). Moreover, with the help of the monotone iterative method, some new existence results on the positive entire k-convex radial solutions of the k-Hessian system with the special non-linearities ψ,φ are given, which improve and extend many previous works.