### AIMS Mathematics

2020, Issue 4: 3840-3850. doi: 10.3934/math.2020249
Research article

# Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space

• Received: 31 December 2019 Accepted: 12 April 2020 Published: 22 April 2020
• MSC : 34B18, 34C23, 35J60

• In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation $\left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) \gt 0, - 1 \lt x \lt 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right.$ where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.

Citation: Zhiqian He, Liangying Miao. Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space[J]. AIMS Mathematics, 2020, 5(4): 3840-3850. doi: 10.3934/math.2020249

### Related Papers:

• In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation $\left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) \gt 0, - 1 \lt x \lt 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right.$ where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.

 [1] S. Y. Cheng, S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104 (1976), 407-419. doi: 10.2307/1970963 [2] R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87 (1982), 131-152. doi: 10.1007/BF01211061 [3] A. Azzollini, On a prescribed mean curvature equation in Lorentz-Minkowski space, J. Math. Pure. Appl., 106 (2016), 1122-1140. doi: 10.1016/j.matpur.2016.04.003 [4] M. Born, L. Infeld, Foundations of the new field theory, Proc. R. Soc. Lond., A, 144 (1934), 425-451. doi: 10.1098/rspa.1934.0059 [5] C. Bereanu, D. de la Fuente, A. Romero, et al. Existence and multiplicity of entire radial space like graphs with prescribed mean curvature function in certain Friedmann-Lemaître- Robertson-Walker space times, Commun. Contemp. Math., 19 (2017), 1-18. doi: 10.1142/S0219199716500061 [6] J. Mawhin, P. J. Torres, Prescribed mean curvature graphs with Neumann boundary conditions in some FLRW spacetimes, J. Differ. Equ., 261 (2016), 7145-7156. doi: 10.1016/j.jde.2016.09.013 [7] M. Born, Modified field equations with a finite radius of the electron, Nature, 132 (1933), 282. [8] G. W. Dai, Global structure of one-sign solutions for problem with mean curvature operator, Nonlinearity, 31 (2018), 5309-5328. doi: 10.1088/1361-6544/aadf43 [9] C. Bereanu, P. Jebelean, P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287. doi: 10.1016/j.jfa.2012.10.010 [10] C. Bereanu, P. Jebelean, P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659. doi: 10.1016/j.jfa.2013.04.006 [11] M. H. Pei, L. B. Wang, Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space, Appl. Math. Lett., 60 (2016), 50-55. doi: 10.1016/j.aml.2016.04.001 [12] I. Coelho, C. Corsato, F. Obersnel, et al. Positive solutions of the Dirichlet problem for the onedimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012) 621-638. [13] X. M. Zhang, M. Q. Feng, Bifurcation diagrams and exact multiplicity of positive solutions of onedimensional prescribed mean curvature equation in Minkowski space, Commun. Contemp. Math., 21 (2019), 1850003. doi: 10.1090/conm/725 [14] R. Y. Ma, Y. Q. Lu, Multiplicity of Positive Solutions for Second Order Nonlinear Dirichlet Problem with One-dimension Minkowski-Curvature Operator, Adv. Nonlinear Stud., 15 (2015), 789-803. doi: 10.1515/ans-2015-0403 [15] K. Deimling, Nonlinear Functional Analysis, Berlin: Springer, 1985. [16] S. C. Hu, H. Y. Wang, Convex Solutions of boundary value problems arising from Monge-Ampère equation, Discrete Cont. Dyn. S., 16 (2006) 705-720. [17] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract cones, Academic press, 1988. [18] P. Candito, R. Livrea, J. Mawhin, Three solutions for a two-point boundary value problem with the prescribed mean curvature equation, Differ. Integral Equ., 28 (2015), 989-1010.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.2 3.4