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

Previous Articles    

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

PDF (PC)

86

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
[1] COELHO I, CORSATO C, OBERSNEL F, et al. Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation[J]. Advanced Nonlinear Studies, 2012, 12(3):621-638.
[2] BEREANU C, JEBELEAN P, TORRES P J. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space[J]. Journal of Functional Analysis, 2013, 264(1):270-287.
[3] BEREANU C, JEBELEAN P, TORRES P J. Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space[J]. Journal of Functional Analysis, 2013, 265(4):644-659.
[4] MA Ruyun, GAO Hongliang, LU Yanqiong. Global structure of radial positive solutions for a prescribed mean curvature problem in a ball[J]. Journal of Functional Analysis, 2016, 270(7):2430-2455.
[5] DAI Guowei. Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space[J]. Calculus of Variations and Partial Differential Equations, 2016, 55(5):1-17.
[6] DAI Guowei. Global bifurcation for problem with mean curvature operator on general domain[J]. Nonlinear Differential Equations and Applications, 2017, 3(24):1-10.
[7] COELHO I, CORSATO C, RIVETTI S. Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball[J]. Topological Methods in Nonlinear Analysis, 2014, 44(1):23-39.
[8] MA Ruyun, CHEN Tinlan, GAO Hongliang. On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2016, 2016(98):1-10.
[9] ZHANG Xuemei, FENG Meiqiang. Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space[J]. Communications in Contemporary Mathematics, 2019, 21(3):1-15.
[10] HUANG Shaoyuan. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application[J]. Communications on Pure and Applied Analysis, 2018, 17(3):1271-1294.
[11] HUANG Shaoyuan. Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications[J]. Journal of Differential Equations, 2018, 264(9):5977- 6011.
[12] GAO Hongliang, XU Jing. Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation[J]. Boundary Value Problems, 2021, 2021(1):1-10.
[13] KORMAN P, LI Y. On the exactness of an S-shaped bifurcation curve[J]. Proceedings of the American Mathematical Society, 1999, 127(4):1011-1020.
[14] HUNG K C, WANG S H. A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem[J]. Journal of Differential Equations, 2011, 251(2):223-237.
[15] KORMAN P, LI Y. Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities[J]. Journal of Differential Equations, 2014, 257(10):3730-3737.
[1] LI Chun-ping, SANG Yan-bin. Multiple solutions of fractional p-q-Laplacian system with sign-changing weight functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(8): 95-102.
[2] SU Xiao-xiao, ZHANG Ya-li. Existence of positive solutions for periodic boundary value problems of secondorder damped difference equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 56-63.
[3] WU Ruo-fei. Existence of solutions for singular fourth-order m-point boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 75-83.
[4] LIU Meng-xue, LI Jie-mei, YAO Yan-yan. Multiplicity of positive solutions for fourth-order boundary value problems with nonlinear boundary conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 84-91.
[5] ZHAO Jiao. Global structure of the set of positive solutions for a class of nonlinear third-order boundary value problems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(10): 104-110.
[6] SUN Yan-yan, LIU Yan-sheng. Existence of positive solutions for singular boundary value problems of Hadamard fractional differential equations in abstract space [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(10): 95-103.
[7] CHEN Rui-peng, LI Xiao-ya. Positive periodic solutions for second-order singular differential equations with damping terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 33-41.
[8] GONG Zeng-tai, GAO Han. Preinvexity of n-dimensional fuzzy number-valued functions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 72-81.
[9] WANG Su-yun, LI Yong-jun. Solvability of nonlinear second-order boundary value problems with nonlinearities which cross the resonance points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 53-56.
[10] . Existence of positive solutions for a class of nonlinear second-order Dirichlet problem [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 64-69.
[11] ZHANG Shen-gui. Multiple solutions of Navier boundary value problem for fourth-order elliptic equation with variable exponents [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 32-37.
[12] YE Fu-mei. Existence results of a resonance problem with derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 25-31.
[13] ZHANG Di, LIU Wen-bin. Existence of solutions for p(t)-Laplacian fractional infinite-point boundary value problems at resonance [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 72-80.
[14] . Periodic solutions for second order singular damped differential equations with a weak singularity [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 84-88.
[15] YAN Dong-liang. Positive solutions of a second order periodic problems with derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 69-75.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd