JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (4): 74-81.doi: 10.6040/j.issn.1671-9352.0.2022.157

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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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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

CLC Number: 

  • O175.8
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