| 带有凸-凹非线性项的平均曲率问题正解的个数 |
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| 引用本文： | 徐晶,高红亮. 带有凸-凹非线性项的平均曲率问题正解的个数[J]. 山东大学学报(理学版), 2023, 58(4): 74-81. DOI: 10.6040/j.issn.1671-9352.0.2022.157 |
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| 作者姓名： | 徐晶 高红亮 |
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| 作者单位： | 兰州交通大学数理学院, 甘肃 兰州 730070 |
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| 基金项目： | 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) |
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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。
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| 关 键 词： | Minkowski-曲率方程 正解的确切个数 分歧曲线 时间映像 |
Number of positive solutions for mean curvature problem with convex-concave nonlinearity |
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| XU Jing,GAO Hong-liang. Number of positive solutions for mean curvature problem with convex-concave nonlinearity[J]. Journal of Shandong University, 2023, 58(4): 74-81. DOI: 10.6040/j.issn.1671-9352.0.2022.157 |
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| Authors: | XU Jing GAO Hong-liang |
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| Affiliation: | School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China |
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| 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. |
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| Keywords: | Minkowski-curvature equation exact multiplicity of positive solution bifurcation curve time map |
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