doi: 10.3934/dcdsb.2018271

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

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

* Corresponding author: Ruyun Ma

Received  January 2018 Revised  May 2018 Published  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.
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, 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. BereanuP. 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. 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.

[4]

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.

[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. CoelhoC. CorsatoF. 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. CoelhoC. 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. CorsatoF. 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. CorsatoF. ObersnelP. 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. 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.

[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. MaH. 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. BereanuP. 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. 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.

[4]

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.

[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. CoelhoC. CorsatoF. 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. CoelhoC. 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. CorsatoF. 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. CorsatoF. ObersnelP. 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. 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.

[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. MaH. 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.

[1]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 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 - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Yong Huang, Lu Xu. Two problems related to prescribed curvature measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1975-1986. doi: 10.3934/dcds.2013.33.1975

[15]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193

[16]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803

[17]

Anca Radulescu. The connected Isentropes conjecture in a space of quartic polynomials. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 139-175. doi: 10.3934/dcds.2007.19.139

[18]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

[19]

Brittany Froese Hamfeldt. Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature. Communications on Pure & Applied Analysis, 2018, 17 (2) : 671-707. doi: 10.3934/cpaa.2018036

[20]

Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (11)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]