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

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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.

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, China

    Li-hong Zhang, Ze-dong Yang & Guo-tao Wang

  2. Shanghai Automotive Wind Tunnel Center, Tongji University, Shanghai, 201804, China

    Mohammad M. Rashidi

Authors
  1. Li-hong Zhang
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  2. Ze-dong Yang
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  3. Guo-tao Wang
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  4. Mohammad M. Rashidi
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Correspondence to Guo-tao Wang.

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Supported by the National Natural Science Foundation of China(11501342,12001344).

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Zhang, Lh., Yang, Zd., Wang, Gt. et al. Classification and existence of positive entire k-convex radial solutions for generalized nonlinear k-Hessian system. Appl. Math. J. Chin. Univ. 36, 564–582 (2021). https://doi.org/10.1007/s11766-021-4363-8

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  • DOI: https://doi.org/10.1007/s11766-021-4363-8

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