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

## Abstract    Full Text(HTML)    Figure/Table    Related pages

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

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

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.