您的位置:山东大学 -> 科技期刊社 -> 《山东大学学报(理学版)》

《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (3): 55-63.doi: 10.6040/j.issn.1671-9352.0.2022.133

• • 上一篇    下一篇

一类奇异k-Hessian方程耦合系统的特征值问题

丁欢欢,何兴玥*   

  1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 发布日期:2023-03-02
  • 作者简介:丁欢欢(1997— ),女,硕士研究生,研究方向为常微分方程与动力系统. E-mail:hding_nwnu@163.com*通信作者简介:何兴玥(1995— ),女,博士研究生,研究方向为常微分方程与动力系统. E-mail:hett199527@163.com
  • 基金资助:
    国家自然科学基金资助项目(11961060);西北师范大学研究生科研资助项目(2021KYZZ01032)

Eigenvalue problem of a coupled system of singular k-Hessian equations

DING Huan-huan, HE Xing-yue*   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2023-03-02

摘要: 考察一类奇异k-Hessian方程耦合系统特征值问题径向解的存在性。通过构造适当的上下解,并利用Schauder不动点定理,证得该问题至少存在一个径向解,并获得该径向解的一些渐近性质。

关键词: k-Hessian方程, Hessian矩阵, 上下解, 奇异性

Abstract: This paper focuses on the existence of radial solutions for the eigenvalue problem of a coupled system of singular k-Hessian equations. By constructing the suitable upper and lower solutions and using the Schauder fixed point theorem, it is proved that at least one radial solution exists in this problem and some asymptotic properties of the radial solution are obtained.

Key words: k-Hessian equation, Hessian matrix, upper and lower solution, singularity

中图分类号: 

  • O175.8
[1] CAFFARELLIL A, NIRENBERG L, SPRUCK J. The Dirichlet problem for nonlinear second order elliptic equations(Ⅲ): functions of the eigenvalues of the Hessian[J]. Acta Mathematica, 1985, 155(3):261-301.
[2] WANG Xujia. The k-Hessian equation[J]. Lecture Notes in Mathematics, 2009, 1977(2):177-252.
[3] LAIR A V, WOOD A W. Large solutions of semilinear elliptic problem[J]. Nonlinear Analysis, 1999, 37(6):805-812.
[4] LAIR A V, WOOD A W. Existence of entire large positive solutions of semilinear elliptic systems[J]. Journal of Differential Equations, 2000, 164(2):380-394.
[5] ZHANG Zhitao, QI Zexin. On a power-type coupled system of Monge-Ampère equations[J]. Topological Methods in Nonlinear Analysis, 2015, 46(2):717-729.
[6] LIU Ronghua, WANG Fanglei, AN Yukun. On radial solutions for Monge-Ampère equations[J]. Turkish Journal of Mathematics, 2018, 42(4):1590-1609.
[7] ZHANG Xuemei, FENG Meiqiang. The existence and asymptotic behavior of boundary blow-up solutions to the k-Hessian equation[J]. Journal of Differential Equations, 2019, 267(8):4626-4672.
[8] ZHANG Xuemei, FENG Meiqiang. Boundary blow-up solutions to the Monge-Ampère equation: sharp conditions and asymptotic behavior[J]. Advances in Nonlinear Analysis, 2020, 9(1):729-744.
[9] HU Shouchuan, WANG Haiyan. Convex solutions of boundary value problems arising from Monge-Ampère equation[J]. Discrete and Continuous Dynamical Systems, 2006, 16(3):705-720.
[10] 梁载涛, 单雪梦. k-Hessian方程径向解的存在性与多解性[J]. 数学物理学报, 2021, 41(1):63-68. LIANG Zaitao, SHAN Xuemeng. Existence and multiplicity of radial solutions of k-Hessian equations[J]. Acta Mathematica Scientia A, 2021, 41(1):63-68.
[11] 段对花, 高承华, 王晶晶. 一类k-Hessian方程爆破解的存在性和不存在性[J]. 山东大学学报(理学版), 2022, 57(3):62-67. DUAN Duihua, GAO Chenghua, WANG Jingjing. Existence and nonexistence of blow-up solutions of k-Hessian equations[J]. Journal of Shandong University(Natural Science), 2022, 57(3):62-67.
[12] FENG Meiqiang, ZHANG Xuemei. A coupled system of k-Hessian equations[J]. Mathematics Methods Applied Scientia, 2019, 2(4):1-18.
[13] WANG Haiyan. Convex solutions of systems arising from Monge-Ampère equations[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2009, 26(8):1-8.
[14] GAO Chenghua, HE Xingyue, RAN Maojun. On a power type coupled system of k-Hessian equations[J]. Quaestiones Mathematicae, 2021, 44(11):1593-1612.
[15] ZHANG Xinguang, XU Jiafa, JIANG Jiqiang, et al. The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general k-Hessian equations[J]. Applied Mathematics Letters, 2020, 102(10):106-124.
[16] ZHANG Xinguang, XU Pengtao, WU Yongyong. The eigenvalue problem of a singular k-Hessian equation[J]. Applied Mathematics Letters, 2022, 124(9):1-9.
[17] JI Xiaohu, BAO Jiguang. Necessary and sufficient conditions on solvability for Hessian inequalities[J]. Proceedings American Mathematical Society, 2010, 138(1):175-188.
[1] 段对花,高承华,王晶晶. 一类k-Hessian方程爆破解的存在性和不存在性[J]. 《山东大学学报(理学版)》, 2022, 57(3): 62-67.
[2] 雷林,李笑丽,何承源. r-H-循环矩阵的性质及其逆的多项式算法[J]. 《山东大学学报(理学版)》, 2021, 56(4): 102-110.
[3] 武若飞. 奇异四阶m-点边值问题解的存在性[J]. 《山东大学学报(理学版)》, 2021, 56(2): 75-83.
[4] 杨晓梅,路艳琼,王瑞. 二阶离散Neumann边值问题的Ambrosetti-Prodi型结果[J]. 《山东大学学报(理学版)》, 2021, 56(2): 64-74.
[5] 苏肖肖. 一类奇异二阶阻尼差分方程周期边值问题正解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(12): 38-45.
[6] 叶芙梅. 带导数项共振问题的可解性[J]. 山东大学学报(理学版), 2018, 53(2): 25-31.
[7] 朱雯雯. 一阶多点边值问题多个解的存在性[J]. 山东大学学报(理学版), 2016, 51(6): 42-48.
[8] 仲秋艳,张兴秋. 含参数及p-Laplacian算子的奇异分数阶微分方程积分边值问题的正解[J]. 山东大学学报(理学版), 2016, 51(6): 78-84.
[9] 朱雯雯. 带参数的一阶周期边值问题正解的存在性及多解性[J]. 山东大学学报(理学版), 2016, 51(12): 36-41.
[10] 吴成明. 二阶奇异耦合系统正周期解的存在性[J]. 山东大学学报(理学版), 2015, 50(10): 81-88.
[11] 马陆一. 非线性二阶Neumann边值问题的Ambrosetti-Prodi型结果[J]. 山东大学学报(理学版), 2015, 50(03): 62-66.
[12] 张秋华, 刘利斌, 周恺. 时滞非局部扩散Lotka-Volterra 竞争系统行波解的存在性[J]. 山东大学学报(理学版), 2015, 50(01): 90-94.
[13] 孙艳梅1,赵增勤2. 一类二阶奇异脉冲微分方程解的存在性[J]. J4, 2013, 48(6): 91-95.
[14] 李凡凡,刘锡平*,智二涛. 分数阶时滞微分方程积分边值问题解的存在性[J]. J4, 2013, 48(12): 24-29.
[15] 运东方,黄淑祥. 一类奇异抛物型偏微分方程的解的存在性和惟一性[J]. J4, 2012, 47(2): 1-7.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!