AIMS Mathematics, 2020, 5(4): 3840-3850. doi: 10.3934/math.2020249

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

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

1 Department of Basic Teaching and Research, Qinghai University, Xining 810016, P. R. China
2 School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, P. R. China

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

References

1. S. Y. Cheng, S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104 (1976), 407-419.    

2. R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87 (1982), 131-152.    

3. A. Azzollini, On a prescribed mean curvature equation in Lorentz-Minkowski space, J. Math. Pure. Appl., 106 (2016), 1122-1140.    

4. M. Born, L. Infeld, Foundations of the new field theory, Proc. R. Soc. Lond., A, 144 (1934), 425-451.    

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.

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.    

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.    

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.    

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.    

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.    

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.

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.

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.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved