数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1414-1426.doi: 10.1007/s10473-022-0409-0

• 论文 • 上一篇    下一篇

THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT

王国涛1, 杨泽栋1, 徐家发2, 张丽红3   

  1. 1. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, China;
    2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China;
    3. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, China
  • 收稿日期:2021-02-12 修回日期:2021-10-26 出版日期:2022-08-25 发布日期:2022-08-23
  • 通讯作者: Jiafa XU,E-mail:xujiafa292@sina.com E-mail:xujiafa292@sina.com
  • 基金资助:
    This work is supported by NSFC (12001344), the Graduate Education and Teaching Innovation Project of Shanxi, China (2021YJJG142), the Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0123), and the Technology Research Foundation of Chongqing Educational Committee (KJQN201900539 and KJQN202000528).

THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT

Guotao WANG1, Zedong YANG1, Jiafa XU2, Lihong ZHANG3   

  1. 1. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, China;
    2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China;
    3. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030031, China
  • Received:2021-02-12 Revised:2021-10-26 Online:2022-08-25 Published:2022-08-23
  • Contact: Jiafa XU,E-mail:xujiafa292@sina.com E-mail:xujiafa292@sina.com
  • Supported by:
    This work is supported by NSFC (12001344), the Graduate Education and Teaching Innovation Project of Shanxi, China (2021YJJG142), the Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0123), and the Technology Research Foundation of Chongqing Educational Committee (KJQN201900539 and KJQN202000528).

摘要: 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.

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

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

中图分类号: 

  • 35L20