November  2020, 13(11): 3047-3071. doi: 10.3934/dcdss.2020118

Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

2. 

Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain

3. 

Departamento de Matemática Aplicada, & Research Unit Modeling Nature (MNat), Universidad de Granada, 18071 Granada, Spain

* Corresponding author

Received  March 2019 Published  November 2020 Early access  October 2019

Fund Project: The first author is supported by NNSF of China (No. 11871129) and Xinghai Youqing funds from Dalian University of Technology, the second one by Spanish MINECO Grant with FEDER funds MTM2016-78807-C2-1-P and the third author by Spanish MINECO Grant with FEDER funds MTM2017-82348-C2-1-P

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime
$ \begin{eqnarray} \text{div} \left(\frac{a\nabla u}{\sqrt{1-a^2\vert \nabla u\vert^2}}\right)+\frac{g(\nabla u, \nabla a)}{\sqrt{1-a^2\vert \nabla u\vert^2}} = \lambda NH \end{eqnarray} $
with
$ 0 $
-Dirichlet boundary condition on the unit ball. According to the behavior of
$ H $
near
$ 0 $
, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs.
Citation: Guowei Dai, Alfonso Romero, Pedro J. Torres. Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3047-3071. doi: 10.3934/dcdss.2020118
References:
[1]

J. A. Aledo, A. Romero and R. M. Rubio, The existence and uniqueness of standard static splitting, Class. Quantum Grav., 32 (2015), 105004, 9 pp. doi: 10.1088/0264-9381/32/10/105004.

[2]

R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.  doi: 10.1007/BF01211061.

[3]

C. BereanuP. Jebelean and 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.

[4]

C. BereanuP. 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), 644-659.  doi: 10.1016/j.jfa.2013.04.006.

[5]

C. BereanuP. Jebelean and J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski spaceÔĢöa variational approach, Adv. Nonlinear Stud., 14 (2014), 315-326.  doi: 10.1515/ans-2014-0204.

[6]

E. Calabi, Examples of Bernstein problems for some nonlinear equations, Global Analysis, Amer. Math. Soc., Providence, R.I., 15 (1970), 223-230. 

[7] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 1998. 
[8]

S.-Y. Cheng and S.-T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104 (1976), 407-419.  doi: 10.2307/1970963.

[9]

C. CorsatoF. ObersnelP. 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), 227-239.  doi: 10.1016/j.jmaa.2013.04.003.

[10]

G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations, 55 (2016), Art. 72, 17 pp. doi: 10.1007/s00526-016-1012-9.

[11]

G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 97, 28 pp. doi: 10.1007/s00526-016-1029-0.

[12]

G. W. Dai, Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.  doi: 10.3934/dcds.2016034.

[13]

G. W. Dai, Global bifurcation for problem with mean curvature operator on general domain, Nonlinear Differential Equations Appl., 24 (2017), Art. 30, 10 pp. doi: 10.1007/s00030-017-0454-x.

[14]

G. W. Dai, Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J., 67 (2018), 2103-2121.  doi: 10.1512/iumj.2018.67.7546.

[15]

G. W. DaiA. Romero and P. J. Torres, Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann-Lemaître-Robertson-Walker spacetimes, J. Differential Equations, 264 (2018), 7242-7269.  doi: 10.1016/j.jde.2018.02.014.

[16]

M. Dajczer, Submanifolds and Isometric Immersions, Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX, 1990.

[17]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.  doi: 10.1512/iumj.1974.23.23087.

[18]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.  doi: 10.1112/S002460930200108X.

[19]

D. Fuente, A. Romero and P. J. Torres, Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner-Nordström spacetimes, Class. Quantum Grav., 32 (2015), 035018, 17 pp. Corrigendum: Class. Quantum Grav., 35 (2018), 059501, 2 pp. doi: 10.1088/1361-6382/aaa5c9.

[20]

E. L. Ince, Ordinary Differential Equation, Dover Publication, New York, 1944.

[21]

J. L. Kazdan, Applications of Partial Differential Equations to Problems in Geometry, Grad. Texts in Math., Springer, 2004.

[22] B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983. 
[23]

R. Osserman, The minimal surface equation, Seminar on Nonlinear Partial Differential Equations, Math. Sci. Res. Inst. Publ., Springer, New York, 2 (1984), 237-259.  doi: 10.1007/978-1-4612-1110-5_13.

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[25]

P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2.

[26]

R. K. Sachs and H. H. Wu, General Relativity for Mathematicians, Graduate Texts in Mathematics, Vol. 48. Springer-Verlag, New York-Heidelberg, 1977.

[27]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140. American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.

[28]

A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39-56.  doi: 10.1007/BF01404755.

[29]

W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.

[30]

G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23. Princeton University Press, Princeton, N. J. 1958.

show all references

References:
[1]

J. A. Aledo, A. Romero and R. M. Rubio, The existence and uniqueness of standard static splitting, Class. Quantum Grav., 32 (2015), 105004, 9 pp. doi: 10.1088/0264-9381/32/10/105004.

[2]

R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.  doi: 10.1007/BF01211061.

[3]

C. BereanuP. Jebelean and 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.

[4]

C. BereanuP. 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), 644-659.  doi: 10.1016/j.jfa.2013.04.006.

[5]

C. BereanuP. Jebelean and J. Mawhin, The Dirichlet problem with mean curvature operator in Minkowski spaceÔĢöa variational approach, Adv. Nonlinear Stud., 14 (2014), 315-326.  doi: 10.1515/ans-2014-0204.

[6]

E. Calabi, Examples of Bernstein problems for some nonlinear equations, Global Analysis, Amer. Math. Soc., Providence, R.I., 15 (1970), 223-230. 

[7] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 1998. 
[8]

S.-Y. Cheng and S.-T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104 (1976), 407-419.  doi: 10.2307/1970963.

[9]

C. CorsatoF. ObersnelP. 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), 227-239.  doi: 10.1016/j.jmaa.2013.04.003.

[10]

G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations, 55 (2016), Art. 72, 17 pp. doi: 10.1007/s00526-016-1012-9.

[11]

G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 97, 28 pp. doi: 10.1007/s00526-016-1029-0.

[12]

G. W. Dai, Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.  doi: 10.3934/dcds.2016034.

[13]

G. W. Dai, Global bifurcation for problem with mean curvature operator on general domain, Nonlinear Differential Equations Appl., 24 (2017), Art. 30, 10 pp. doi: 10.1007/s00030-017-0454-x.

[14]

G. W. Dai, Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J., 67 (2018), 2103-2121.  doi: 10.1512/iumj.2018.67.7546.

[15]

G. W. DaiA. Romero and P. J. Torres, Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann-Lemaître-Robertson-Walker spacetimes, J. Differential Equations, 264 (2018), 7242-7269.  doi: 10.1016/j.jde.2018.02.014.

[16]

M. Dajczer, Submanifolds and Isometric Immersions, Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX, 1990.

[17]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.  doi: 10.1512/iumj.1974.23.23087.

[18]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.  doi: 10.1112/S002460930200108X.

[19]

D. Fuente, A. Romero and P. J. Torres, Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner-Nordström spacetimes, Class. Quantum Grav., 32 (2015), 035018, 17 pp. Corrigendum: Class. Quantum Grav., 35 (2018), 059501, 2 pp. doi: 10.1088/1361-6382/aaa5c9.

[20]

E. L. Ince, Ordinary Differential Equation, Dover Publication, New York, 1944.

[21]

J. L. Kazdan, Applications of Partial Differential Equations to Problems in Geometry, Grad. Texts in Math., Springer, 2004.

[22] B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983. 
[23]

R. Osserman, The minimal surface equation, Seminar on Nonlinear Partial Differential Equations, Math. Sci. Res. Inst. Publ., Springer, New York, 2 (1984), 237-259.  doi: 10.1007/978-1-4612-1110-5_13.

[24]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.

[25]

P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.  doi: 10.1016/0022-0396(73)90061-2.

[26]

R. K. Sachs and H. H. Wu, General Relativity for Mathematicians, Graduate Texts in Mathematics, Vol. 48. Springer-Verlag, New York-Heidelberg, 1977.

[27]

G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140. American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.

[28]

A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39-56.  doi: 10.1007/BF01404755.

[29]

W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0601-9.

[30]

G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23. Princeton University Press, Princeton, N. J. 1958.

Figure 1.  Bifurcation diagrams of Theorem 1.1
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