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June  2019, 24(6): 2701-2718. doi: 10.3934/dcdsb.2018271

Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space

Ruyun Ma , and  Man Xu

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Ruyun Ma

Received  January 2018 Revised  May 2018 Published  June 2019 Early access  October 2018

Fund Project: The first author is supported by NSF grant NSFC (No.11671322).

In this paper we study global bifurcation phenomena for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space
$\left\{ \begin{array}{l} -\text{div}\big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\big) = λ f(x,u,\nabla u)\ \ \ \ \ \ & \text{in}\ Ω,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{on}\ \partial Ω.\\\end{array} \right.$
Here
$Ω$
is a bounded regular domain in
$\mathbb{R}^N$
, the function
$f$
satisfies the Carathéodory conditions, and
$f$
is either superlinear or sublinear in
$u$
at
$0$
. The proof of our main results are based upon bifurcation techniques.
Keywords: Prescribed mean curvature equations, Lorentz-Minkowski space, positive solution, bifurcation, connected component.
Mathematics Subject Classification: Primary: 35J25; Secondary: 47H11, 47J10, 34B18.
Citation: Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
References:
[1]

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

[2]

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

[3]

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

[4]

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), 270-287.  doi: 10.1016/j.jfa.2012.10.010.

[5]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[6]

G. Chen, Introduction to Sobelev Spaces, Science press, Beijing, 2013.

[7]

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

[8]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Fundamental Principles of Mathematical Science, 251, Springer-Verlag, New York-Berlin, 1982.

[9]

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), 621-638.  doi: 10.1515/ans-2012-0310.

[10]

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), 23-39.  doi: 10.12775/TMNA.2014.034.

[11]

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), 113-134.  doi: 10.1515/gmj-2016-0078.

[12]

C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, Discrete Contin. Dyn. Syst. Suppl., 2013 (2013), 159-169.  doi: 10.3934/proc.2013.2013.159.

[13]

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

[14]

C. Gerhardt, H-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553. 

[15]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.  doi: 10.1080/03605308008820162.

[16]

H. Kielhöfer, Bifurcation Theory. An Introduction with Applications to PDEs, 156, Applied Mathematical Sciences, Springer-Verlag, New York, 2004.

[17]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[18]

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), 279-290.  doi: 10.12775/TMNA.2009.043.

[19]

R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364-4376.  doi: 10.1016/j.na.2009.02.113.

[20]

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), 2430-2455.  doi: 10.1016/j.jfa.2016.01.020.

[21]

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

[22]

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

[23]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.

[24]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems, Translated from the German by Peter R. Wadsack., Springer-Verlag, New York, 1986.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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), 270-287.  doi: 10.1016/j.jfa.2012.10.010.

[5]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[6]

G. Chen, Introduction to Sobelev Spaces, Science press, Beijing, 2013.

[7]

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

[8]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Fundamental Principles of Mathematical Science, 251, Springer-Verlag, New York-Berlin, 1982.

[9]

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), 621-638.  doi: 10.1515/ans-2012-0310.

[10]

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), 23-39.  doi: 10.12775/TMNA.2014.034.

[11]

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), 113-134.  doi: 10.1515/gmj-2016-0078.

[12]

C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, Discrete Contin. Dyn. Syst. Suppl., 2013 (2013), 159-169.  doi: 10.3934/proc.2013.2013.159.

[13]

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

[14]

C. Gerhardt, H-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553. 

[15]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.  doi: 10.1080/03605308008820162.

[16]

H. Kielhöfer, Bifurcation Theory. An Introduction with Applications to PDEs, 156, Applied Mathematical Sciences, Springer-Verlag, New York, 2004.

[17]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[18]

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), 279-290.  doi: 10.12775/TMNA.2009.043.

[19]

R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions, Nonlinear Anal., 71 (2009), 4364-4376.  doi: 10.1016/j.na.2009.02.113.

[20]

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), 2430-2455.  doi: 10.1016/j.jfa.2016.01.020.

[21]

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

[22]

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

[23]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.

[24]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems, Translated from the German by Peter R. Wadsack., Springer-Verlag, New York, 1986.

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