一类k-Hessian方程解的存在性和渐近稳定性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1357-1371.

一类k-Hessian方程解的存在性和渐近稳定性

张丽红¹(),杨泽栋¹(),王国涛^1,^2,*(),BaleanuDumitru^3,⁴()

¹ 山西师范大学数学与计算机科学学院山西临汾 041004
² Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia Jeddah 21589
³ Department of Mathematics, Faculty of Art and Sciences, Cankaya University, Turkey, Ankara Balgat 06530
⁴ Institute of Space Sciences, Magurele-Bucharest, Romania

收稿日期:2020-11-07 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 王国涛 E-mail:zhanglih149@126.com;yangzd1229@163.com;wgt2512@163.com;dumitru@cankaya.edu.tr
作者简介:张丽红, E-mail: zhanglih149@126.com|杨泽栋, E-mail: yangzd1229@163.com|Baleanu Dumitru, E-mail: dumitru@cankaya.edu.tr
基金资助:
国家自然科学基金(11501342);国家自然科学基金(12001344);山西省研究生教育创新项目基金(2020SY337)

Existence and Asymptotic Behavior of Solutions of a Class of k-Hessian Equation

Lihong Zhang¹(),Zedong Yang¹(),Guotao Wang^1,^2,*(),Dumitru Baleanu^3,⁴()

¹ School of Mathematics and Computer Science, Shanxi Normal University, Shanxi Linfen 041004

Received:2020-11-07 Online:2021-10-26 Published:2021-10-08
Contact: Guotao Wang E-mail:zhanglih149@126.com;yangzd1229@163.com;wgt2512@163.com;dumitru@cankaya.edu.tr
Supported by:
the NSFC(11501342);the NSFC(12001344);the Graduate Education Innovation Program Fund of Shanxi(2020SY337)

摘要/Abstract

摘要：

该文考虑了边界爆破$k$-Hessian问题 $S_{k}\left(\lambda\left(D^{2}z\right)\right)=b(x)f(z)，~~x\in\Omega，~~z\mid_{\partial\Omega}=+\infty，$ 其中，$\Omega\subset\mathbb{R}^{N}$是一个严格凸的光滑有界区域.文章通过单调迭代方法、上下解方法和Karamata正则变化理论得到了$k$-Hessian方程径向对称正解的存在性和严格凸的爆破正解的边界渐近行为.

关键词: 渐近稳定性, k-Hessian方程, 径向正解, Keller-Osserman条件, Karamata正则变化理论

Abstract:

In this paper, we consider the following boundary blow-up $k$-Hessian problem $S_{k}\left(\lambda\left(D^{2} z\right)\right)=b(x) f(z),~~x \in \Omega,~~z\mid_{\partial\Omega}=+\infty,$ where $\Omega \subset \mathbb{R} ^{N}$ is a smooth, bounded, strictly convex domain. We are concerned with the existence of the radially symmetric positive solutions of the $k$-Hessian equation and obtain new boundary asymptotic behavior of strictly convex blow-up positive solutions of the $k$-Hessian equation. Our approach mainly relies on the monotone iterative method, the upper and lower solution method and Karamata regular variation theory.

Key words: Asymptotic behavior, k-Hessian equation, Positive radial solution, Keller-Osserman condition, Karamata regular variation theory

中图分类号:

O175

张丽红,杨泽栋,王国涛,BaleanuDumitru. 一类k-Hessian方程解的存在性和渐近稳定性[J]. 数学物理学报, 2021, 41(5): 1357-1371.

Lihong Zhang,Zedong Yang,Guotao Wang,Dumitru Baleanu. Existence and Asymptotic Behavior of Solutions of a Class of k-Hessian Equation[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1357-1371.

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