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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 |
$\left\{ \begin{array}{l} -\text{div}\big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\big) = λ f(x,u,\nabla u)\ \ \ \ \ \ & \text{in}\ Ω,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{on}\ \partial Ω.\\\end{array} \right.$ |
$Ω$ |
$\mathbb{R}^N$ |
$f$ |
$f$ |
$u$ |
$0$ |
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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