American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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. Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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. Google Scholar

[1]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169
[2]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[3]

Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155
[4]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[5]

Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367
[6]

Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
[7]

Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142
[8]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[9]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[10]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150
[11]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[12]

Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013
[13]

Franco Obersnel, Pierpaolo Omari. On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation. Conference Publications, 2011, 2011 (Special) : 1138-1147. doi: 10.3934/proc.2011.2011.1138
[14]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[15]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463
[16]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[17]

Guowei Dai, Alfonso Romero, Pedro J. Torres. Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3047-3071. doi: 10.3934/dcdss.2020118
[18]

Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010
[19]

Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076
[20]

Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (157)
  • HTML views (561)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences