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

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

Chenghua GAO, Xingyue HE, Jingjing WANG   

  1. 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   

  1. 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: 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