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

Abstract

In this paper, we consider the following generalized nonlinear k-Hessian system

$$\left\{ {\matrix{{{\cal G}\left( {S_k^{{1 \over k}}(\lambda ({D^2}{z_1}))} \right)S_k^{{1 \over k}}(\lambda ({D^2}{z_1})) = \varphi (\left| x \right|,{z_1},{z_2}),\,\,\,x \in {\mathbb{R}^N},} \cr {{\cal G}\left( {S_k^{{1 \over k}}(\lambda ({D^2}{z_2}))} \right)S_k^{{1 \over k}}(\lambda ({D^2}{z_2})) = \varphi (\left| x \right|,{z_1},{z_2}),\,\,\,x \in {\mathbb{R}^N},} \cr } \,} \right.$$

where \({\cal G}\) is a nonlinear operator and S _k (λ(D ² z)) 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∣, z ₁, z ₂) = b(∣x∣)φ(z ₁, z ₂) and ψ(∣x∣, z ₁, z₂) = h(∣x∣)ψ(z ₁). 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.