Acknowledgement
Supported by : NSFC
References
- L. J. Alias and B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc. 33 (2001), no. 4, 454-458. https://doi.org/10.1017/S0024609301008220
- R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), no. 1, 131-152. https://doi.org/10.1007/BF01211061
- C. Bereanu, P. Jebelean, and P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal. 264 (2013), no. 1, 270-287. https://doi.org/10.1016/j.jfa.2012.10.010
- C. Bereanu, P. Jebelean, and P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), no. 4, 644-659. https://doi.org/10.1016/j.jfa.2013.04.006
-
C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular
${\phi}$ -Laplacian, J. Differential Equations 243 (2007), no. 2, 536-557. https://doi.org/10.1016/j.jde.2007.05.014 - S. Cano-Casanova, J. Lopez-Gomez, and K. Takimoto, A quasilinear parabolic perturbation of the linear heat equation, J. Differential Equations 252 (2012), no. 1, 323-343. https://doi.org/10.1016/j.jde.2011.09.018
- S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407-419. https://doi.org/10.2307/1970963
- I. Coelho, C. Corsato, F. Obersnel, and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud. 12 (2012), no. 3, 621-638. https://doi.org/10.1515/ans-2012-0310
- I. Coelho, C. Corsato, and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 23-39.
- C. Corsato, F. Obersnel, and P. Omari, The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space, Georgian Math. J. 24 (2017), no. 1, 113-134. https://doi.org/10.1515/gmj-2016-0078
- C. Corsato, F. Obersnel, P. Omari, and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl. 405 (2013), no. 1, 227-239. https://doi.org/10.1016/j.jmaa.2013.04.003
- G. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 72, 17 pp.
- K.-C. Hung, S.-H. Wang, and C.-H. Yu, Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl. 392 (2012), no. 1, 40-54. https://doi.org/10.1016/j.jmaa.2012.02.036
- P. Korman, A global solution curve for a class of periodic problems, including the relativistic pendulum, Appl. Anal. 93 (2014), no. 1, 124-136. https://doi.org/10.1080/00036811.2012.762088
- H. J. Li and C. C. Yeh, Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1193-1204. https://doi.org/10.1017/S0308210500030468
- Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations 260 (2016), no. 12, 8358-8387. https://doi.org/10.1016/j.jde.2016.02.021
- R. Lopez, Stationary surfaces in Lorentz-Minkowski space, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 5, 1067-1096. https://doi.org/10.1017/S0308210507000273
- R. Ma and Y. An, Global structure of positive solutions for superlinear second order m-point boundary value problems, Topol. Methods Nonlinear Anal. 34 (2009), no. 2, 279-290. https://doi.org/10.12775/TMNA.2009.043
- R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal. 71 (2009), no. 10, 4364-4376. https://doi.org/10.1016/j.na.2009.02.113
- R. Ma, T. Chen, and H. Gao, On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 98, 10 pp.
- R. Ma, H. Gao, and Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, J. Funct. Anal. 270 (2016), no. 7, 2430-2455. https://doi.org/10.1016/j.jfa.2016.01.020
- R. Ma and Y. Lu, Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud. 15 (2015), no. 4, 789-803. https://doi.org/10.1515/ans-2015-0403
- H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions, J. Differential Equations 254 (2013), no. 3, 1464-1499. https://doi.org/10.1016/j.jde.2012.10.025
- P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487-513. https://doi.org/10.1016/0022-1236(71)90030-9
- T. Shibata, S-shaped bifurcation curves for nonlinear two-parameter problems, Nonlinear Anal. 95 (2014), 796-808. https://doi.org/10.1016/j.na.2013.10.015
- I. Sim and S. Tanaka, Three positive solutions for one-dimensional p-Laplacian problem with sign-changing weight, Appl. Math. Lett. 49 (2015), 42-50. https://doi.org/10.1016/j.aml.2015.04.007
- A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39-56. https://doi.org/10.1007/BF01404755
- W. Walter, Ordinary Differential Equations, translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Springer-Verlag, New York, 1998.
- S.-H. Wang and T.-S. Yeh, Exact multiplicity of solutions and S-shaped bifurcation curves for the p-Laplacian perturbed Gelfand problem in one space variable, J. Math. Anal. Appl. 342 (2008), no. 2, 1175-1191. https://doi.org/10.1016/j.jmaa.2007.12.026
- X. Xu, B. Qin and W. Li, S-shaped bifurcation curve for a nonlocal boundary value problem, J. Math. Anal. Appl. 450 (2017), no. 1, 48-62. https://doi.org/10.1016/j.jmaa.2016.12.073