JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (3): 55-63.doi: 10.6040/j.issn.1671-9352.0.2022.133

Previous Articles     Next Articles

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

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

CLC Number: 

  • 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] DUAN Dui-hua, GAO Cheng-hua, WANG Jing-jing. Existence and nonexistence of blow-up solutions for a general k-Hessian equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(3): 62-67.
[2] LEI Lin, LI Xiao-li, HE Cheng-yuan. Properties for r-H-circulant matrices and polynomial algorithm of their inverse [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(4): 102-110.
[3] WU Ruo-fei. Existence of solutions for singular fourth-order m-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 75-83.
[4] CHEN Rui-peng, LI Xiao-ya. Positive periodic solutions for second-order singular differential equations with damping terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 33-41.
[5] KONG Yi-ting, WANG Tong-ke. The steepest descent method for Fourier integrals involving algebraic and logarithmic singular factors [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 50-55.
[6] ZHONG Qiu-yan, ZHANG Xing-qiu. Positive solutions for some singular fractional differential equation integral boundary value problems with p-Laplacian and a parameter [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 78-84.
[7] ZHU Wen-wen. Existence of multiple of solutions of first order multi-point boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 42-48.
[8] ZHU Wen-wen. Existence and multiplicity of positive solutions of first order periodic boundary value problems with parameter [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 36-41.
[9] WU Cheng-ming. Existence of positive periodic solutions for second order singular coupled systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(10): 81-88.
[10] MA Lu-yi. The Ambrosetti-Prodi type results of the nonlinear second-order Neumann boundary value problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 62-66.
[11] SUN Yan-mei1, ZHAO Zeng-qin2. Existence of solutions for a class of second order  singular impulsive differential equations [J]. J4, 2013, 48(6): 91-95.
[12] LI Fan-fan, LIU Xi-ping*, ZHI Er-tao. Existence of a solution for the boundary value problem of
fractional differential equation with delay
[J]. J4, 2013, 48(12): 24-29.
[13] ZHANG Xiang, HUANG Shu-xiang. Monotone iterative technique, existence and uniqueness results for the nonlinear fractional reaction-diffusion equation [J]. J4, 2011, 46(2): 9-14.
[14] CAI Jing-jing1, LIU Gui-long2. Positive solutions for singular third order two-point boundary value problem in abstract space [J]. J4, 2010, 45(8): 62-65.
[15] WANG Xin-hua, ZHANG Xing-qiu. Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions [J]. J4, 2010, 45(8): 76-80.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!