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

doi:10.1007/s10473-022-0409-0

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

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. 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. 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/Abstract

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

王国涛, 杨泽栋, 徐家发, 张丽红. THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT[J]. 数学物理学报(英文版), 2022, 42(4): 1414-1426.

Guotao WANG, Zedong YANG, Jiafa XU, Lihong ZHANG. THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1414-1426.

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

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

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