THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM*

doi:10.1007/s10473-023-0618-1

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2615-2628.doi: 10.1007/s10473-023-0618-1

THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*

Chenghua GAO^†, Xingyue HE, Jingjing WANG

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

收稿日期:2022-04-27 修回日期:2023-05-30 发布日期:2023-12-08
通讯作者: †Chenghua GAO, E-mail: gaokuguo@163.com;
作者简介:Xingyue HE, E-mail: hett199527@163.com; Jingjing WANG, E-mail: WJJ950712@163.com
基金资助:
This work was supported by the National Natural Science Foundation of China (11961060) and the Graduate Research Support of Northwest Normal University (2021KYZZ01032).

THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*

Chenghua GAO^†, Xingyue HE, Jingjing WANG

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received:2022-04-27 Revised:2023-05-30 Published:2023-12-08
Contact: †Chenghua GAO, E-mail: gaokuguo@163.com;
About author:Xingyue HE, E-mail: hett199527@163.com; Jingjing WANG, E-mail: WJJ950712@163.com
Supported by:
This work was supported by the National Natural Science Foundation of China (11961060) and the Graduate Research Support of Northwest Normal University (2021KYZZ01032).

摘要/Abstract

摘要： In this paper, we focus on the following coupled system of $k$-Hessian equations:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{equation*} \left\{\begin{aligned}&S_k(\lambda(D^2u))=f_1(|x|,-v)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&S_k(\lambda(D^2v))=f_2(|x|,-u)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&u=v=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm on}\ \partial B.\end{aligned}\right.\end{equation*}$
Here B is a unit ball with center 0 and $f_i (i=1,2)$ are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearities $f_1$ and $f_2$, which are more flexible than the existing conditions for the k-Hessian systems (equations), several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.

关键词: system of k-Hessian equations, k-convex solutions, existence, multiplicity, fixed-point theorem

Abstract: In this paper, we focus on the following coupled system of $k$-Hessian equations:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{equation*} \left\{\begin{aligned}&S_k(\lambda(D^2u))=f_1(|x|,-v)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&S_k(\lambda(D^2v))=f_2(|x|,-u)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&u=v=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm on}\ \partial B.\end{aligned}\right.\end{equation*}$
Here B is a unit ball with center 0 and $f_i (i=1,2)$ are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearities $f_1$ and $f_2$, which are more flexible than the existing conditions for the k-Hessian systems (equations), several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.

Key words: system of k-Hessian equations, k-convex solutions, existence, multiplicity, fixed-point theorem

中图分类号:

35J96

Chenghua GAO, Xingyue HE, Jingjing WANG. THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*[J]. 数学物理学报(英文版), 2023, 43(6): 2615-2628.

Chenghua GAO, Xingyue HE, Jingjing WANG. THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2615-2628.

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THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*

THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*

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