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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (4): 74-81.doi: 10.6040/j.issn.1671-9352.0.2022.157

• • 上一篇    

带有凸-凹非线性项的平均曲率问题正解的个数

徐晶,高红亮*   

  1. 兰州交通大学数理学院, 甘肃 兰州 730070
  • 发布日期:2023-03-27
  • 作者简介:徐晶(1997— ),女,硕士研究生,研究方向为微分方程边值问题. E-mail:XJ18734569807@163.com*通信作者简介:高红亮(1985— ),男,副教授,研究方向为分歧理论及常微分方程边值问题. E-mail:gaohongliang101@163.com
  • 基金资助:
    国家自然科学基金资助项目(11801243);甘肃省高等学校青年博士基金项目(2022QB-056);兰州交通大学青年基金资助项目(2017012)

Number of positive solutions for mean curvature problem with convex-concave nonlinearity

XU Jing, GAO Hong-liang*   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Published:2023-03-27

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摘要： 考虑一维Minkowski空间中给定平均曲率问题{-((u')/((1-u'2)1/2))'=λf(u), x∈(-L,L),u(-L)=0=u(L)正解的确切个数及其分歧曲线,其中参数λ>0,L>0, f∈C1［0,∞)∩C2(0,∞), f(0)=0, f(u)>0,u∈(0,L)且f在(0,L)上为凸-凹函数。通过详细的时间映像分析,在两种不同的情况下,根据λ的不同范围,获得了该问题没有正解,恰有一个或两个正解的结果。

关键词: Minkowski-曲率方程, 正解的确切个数, 分歧曲线, 时间映像

Abstract: This paper considers the exact multiplicity and bifurcation curves of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of{-((u')/((1-u'2)1/2))'=λf(u), x∈(-L,L),u(-L)=0=u(L)where λ,L are positive parameters, f∈C1［0,∞)∩C2(0,∞)satisfies f(0)=0, and f(u)>0,u∈(0,L)and f is convex-concave on (0,L). By using a detailed analysis of the time map, it is obtained that the above problem has zero, exactly one or exactly two positive solutions according to different ranges of λ in two different cases.

Key words: Minkowski-curvature equation, exact multiplicity of positive solution, bifurcation curve, time map

中图分类号: 

  • O175.8
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