带有凹非线性项的平均曲率半正问题正解的确切个数

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1341-1349.

带有凹非线性项的平均曲率半正问题正解的确切个数

李晓东^*(),高红亮(),徐晶

兰州交通大学数学系兰州 730070

收稿日期:2022-10-09 修回日期:2023-04-10 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 李晓东 E-mail:LXD5775@163.com;gaohongliang101@163.com
作者简介:高红亮，Email: gaohongliang101@163.com
基金资助:
国家自然科学基金(11801243);国家自然科学基金(11961039);甘肃省高等学校青年博士基金项目(2022QB-056);兰州交通大学青年学者科学基金(2017012)

Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity

Li Xiaodong^*(),Gao Hongliang(),Xu Jing

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070

Received:2022-10-09 Revised:2023-04-10 Online:2023-10-26 Published:2023-08-09
Contact: Xiaodong Li E-mail:LXD5775@163.com;gaohongliang101@163.com
Supported by:
NSFC(11801243);NSFC(11961039);Gansu Province Colleges and Universities Young Doctor fund project(2022QB-056);Young Scholars Science Foundation of Lanzhou Jiaotong University(2017012)

摘要/Abstract

摘要：

该文研究了一维 Minkowski 空间中给定平均曲率问题

$ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L) \end{array} \right. $

正解的确切个数及分歧图, 其中 $\lambda>0$ 为参数, $L>0$ 为常数, $f\in C^{2}([0,\infty), \mathbb{R})$ 满足 $f(0)<0$, 并且对于 $0, $f''(u)<0$. 基于时间映像原理, 讨论了两种情形, 得到了该问题根据 $\lambda$ 的取值范围不同, 分别有零解, 一个解和两个解.

关键词: Minkowski 空间, 半正, 正解, 时间映像, 确切个数

Abstract:

In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of

$ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L), \end{array} \right. $

where $\lambda>0$ is a bifurcation parameter and $L>0$ is an evolution parameters, $f\in C^{2}([0,\infty), \mathbb{R})$ satisfies $f(0)<0$ and $f$ is concave for $0. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of $\lambda$. The arguments are based upon a detailed analysis of the time map.

Key words: Minkowski space, Semipositone, Positive solution, Time map, Exact multiplicity

中图分类号:

O175.8

李晓东, 高红亮, 徐晶. 带有凹非线性项的平均曲率半正问题正解的确切个数[J]. 数学物理学报, 2023, 43(5): 1341-1349.

Li Xiaodong, Gao Hongliang, Xu Jing. Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1341-1349.

图/表 3

参考文献 13

[1]	Bratu G. Sur les equation integrals non linéarires. Bull Soc Math France, 1914, 42: 113-142
[2]	Corsato C. Mathematical Analysis of Some Differential Models Involving the Euclidean or the Minkowski Mean Curvature Operator. Trieste: University of Trieste, 2015
[3]	Gao H L, Xu J. Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation. Boundary Value Problems, 2021, Article number 81
[4]	Gelfand I M. Some problems in the theory of quasilinear equations. American Mathematical Society Translations, 1963, 29(2): 295-381
[5]	Huang S Y. Global bifurcation diagrams for Liouville-Bratu-Gelfand problem with Minkowski-curvature operator. Journal of Dynamics and Differential Equations, 2021, 34: 2157-2172 doi: 10.1007/s10884-021-09982-4
[6]	Huang S Y. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17(3): 1271-1294 doi: 10.3934/cpaa.2018061
[7]	Huang S Y. Classifification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications. Journal of Differential Equations, 2018, 264(9): 5977-6011 doi: 10.1016/j.jde.2018.01.021
[8]	Huang S Y. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18(6): 3267-3284 doi: 10.3934/cpaa.2019147
[9]	Huang S Y. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems-A, 2019, 39(6): 3443-3462
[10]	Huang S Y, Hwang M S. Bifurcation curves of positive solutions for the Minkowski-curvature problem with cubic nonlinearity. Electronic Journal of Qualitative Theory of Differential Equations, 2021, Article number 41
[11]	Hutten E H. Relativistic (non-linear) Oscillator. Nature, 1965, 205(4974): 892
[12]	Walter G. Classicai Mechanics-point Particles and Relativity. New York: Springer, 2004: 1-485
[13]	Zhang X M, Feng M Q. Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space. Communications in Contemporary Mathematics, 2019, 21(3): 111-137

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