January  2020, 40(1): 133-151. doi: 10.3934/dcds.2020006

Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space

Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan 300223, Timişoara, Romania

* Corresponding author: Petru Jebelean

Received  November 2018 Published  October 2019

We deal with a multiparameter Dirichlet system having the form
$ \begin{equation*} \left\{ \begin{array}{ll} -\mathcal M(u) = \lambda_1f_1(u,v), & \hbox{in $\Omega$},\\ -\mathcal M(v) = \lambda_2f_2(u,v), & \hbox{in $\Omega$},\\ u|_{\partial\Omega} = 0 = v|_{\partial\Omega}, \end{array} \right. \end{equation*} $
where
$ \mathcal M $
stands for the mean curvature operator in Minkowski space
$ \mathcal M(u) = \mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right), $
$ \Omega $
is a general bounded regular domain in
$ \mathbb{R}^N $
and the continuous functions
$ f_1,f_2 $
satisfy some sign and quasi-monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For such a system we show the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one
$ \mathcal{O}_0 $
and a closed one
$ \mathcal{F} $
, such that the system has zero or at least one strictly positive solution, according to
$ (\lambda_1, \lambda_2)\in \mathcal{O}_0 $
or
$ (\lambda_1, \lambda_2)\in \mathcal{F} $
. Moreover, we show that inside of
$ \mathcal{F} $
there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. Our approach relies on a lower and upper solutions method - which we develop here, together with topological degree type arguments. In a sense, our results extend to non-radial systems some recent existence/non-existence and multiplicity results obtained in the radial case.
Citation: Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
References:
[1]

L. J. Alías and B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc., 33 (2001), 454-458.  doi: 10.1017/S0024609301008220.  Google Scholar

[2]

L. J. AlíasA. Romero and M. Sánchez, Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems, Tôhoku Math. J., 49 (1997), 337-345.  doi: 10.2748/tmj/1178225107.  Google Scholar

[3]

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.  Google Scholar

[4]

C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for systems involving mean curvature operators in Euclidean and Minkowski spaces, in Mathematical Models in Engineering, Biology and Medicine, AIP Conf. Proc., A. Cabada, E. Liz and J. J. Nieto (eds.), Amer. Inst. Phys., Melville, 1124 (2009), 50–59.  Google Scholar

[5]

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.  Google Scholar

[6]

C. BereanuP. Jebelean and J. Mawhin, Corrigendum to: "The Dirichlet problem with mean curvature operator in Minkowski space - a variational approach" [Adv. Nonlinear Stud., 14 (2014), 315–326], Adv. Nonlinear Stud., 16 (2016), 173-174.  doi: 10.1515/ans-2015-5030.  Google Scholar

[7]

C. BereanuP. Jebelean and P. J. Torres, Multiple positive radial solutions for 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

[8]

E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math., 15 (1970), 223-230.   Google Scholar

[9]

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.  Google Scholar

[10] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, 2009.   Google Scholar
[11]

Y. Choquet-Bruhat, A. E. Fischer and J. E. Marsden, Maximal Hypersurfaces and Positivity of Mass, Proc. of the Enrico Fermi Summer School of the Italian Physical Society, J. Ehlers (Ed.), North-Holland, 1979. Google Scholar

[12]

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.  Google Scholar

[13]

C. CorsatoF. Obersnel and P. Omari, The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorenz-Minkowski space, Georgian Math. J., 24 (2017), 113-134.  doi: 10.1515/gmj-2016-0078.  Google Scholar

[14]

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., 2013 (2013), 159-169.  doi: 10.3934/proc.2013.2013.159.  Google Scholar

[15]

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.  Google Scholar

[16]

D. Gurban and P. Jebelean, Positive radial solutions for systems with mean curvature operator in Minkowski space, Rend. Instit. Mat. Univ. Trieste, 49 (2017), 245-264.  doi: 10.13137/2464-8728/16215.  Google Scholar

[17]

D. Gurban and P. Jebelean, Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities, J. Differential Equations, 266 (2019), 5377-5396.  doi: 10.1016/j.jde.2018.10.030.  Google Scholar

[18]

D. GurbanP. Jebelean and C. Şerban, Nontrivial solutions for potential systems involving the mean curvature operator in Minkowski space, Adv. Nonlinear Stud., 17 (2017), 769-780.  doi: 10.1515/ans-2016-6025.  Google Scholar

[19]

Y.-H. Lee, Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain, Nonlinear Anal., 47 (2001), 3649-3660.  doi: 10.1016/S0362-546X(01)00485-0.  Google Scholar

[20]

R. MaT. Chen and H. Gao, On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator, Electron. J. Qual. Theory Differ. Equ., 98 (2016), 1-10.  doi: 10.14232/ejqtde.2016.1.98.  Google Scholar

[21]

J. E. Marsden and F. J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep., 66 (1980), 109-139.  doi: 10.1016/0370-1573(80)90154-4.  Google Scholar

[22]

R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., 65 (1979), 45-76.  doi: 10.1007/BF01940959.  Google Scholar

[23]

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

show all references

References:
[1]

L. J. Alías and B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc., 33 (2001), 454-458.  doi: 10.1017/S0024609301008220.  Google Scholar

[2]

L. J. AlíasA. Romero and M. Sánchez, Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems, Tôhoku Math. J., 49 (1997), 337-345.  doi: 10.2748/tmj/1178225107.  Google Scholar

[3]

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.  Google Scholar

[4]

C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for systems involving mean curvature operators in Euclidean and Minkowski spaces, in Mathematical Models in Engineering, Biology and Medicine, AIP Conf. Proc., A. Cabada, E. Liz and J. J. Nieto (eds.), Amer. Inst. Phys., Melville, 1124 (2009), 50–59.  Google Scholar

[5]

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.  Google Scholar

[6]

C. BereanuP. Jebelean and J. Mawhin, Corrigendum to: "The Dirichlet problem with mean curvature operator in Minkowski space - a variational approach" [Adv. Nonlinear Stud., 14 (2014), 315–326], Adv. Nonlinear Stud., 16 (2016), 173-174.  doi: 10.1515/ans-2015-5030.  Google Scholar

[7]

C. BereanuP. Jebelean and P. J. Torres, Multiple positive radial solutions for 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

[8]

E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math., 15 (1970), 223-230.   Google Scholar

[9]

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.  Google Scholar

[10] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford University Press, 2009.   Google Scholar
[11]

Y. Choquet-Bruhat, A. E. Fischer and J. E. Marsden, Maximal Hypersurfaces and Positivity of Mass, Proc. of the Enrico Fermi Summer School of the Italian Physical Society, J. Ehlers (Ed.), North-Holland, 1979. Google Scholar

[12]

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.  Google Scholar

[13]

C. CorsatoF. Obersnel and P. Omari, The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorenz-Minkowski space, Georgian Math. J., 24 (2017), 113-134.  doi: 10.1515/gmj-2016-0078.  Google Scholar

[14]

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., 2013 (2013), 159-169.  doi: 10.3934/proc.2013.2013.159.  Google Scholar

[15]

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.  Google Scholar

[16]

D. Gurban and P. Jebelean, Positive radial solutions for systems with mean curvature operator in Minkowski space, Rend. Instit. Mat. Univ. Trieste, 49 (2017), 245-264.  doi: 10.13137/2464-8728/16215.  Google Scholar

[17]

D. Gurban and P. Jebelean, Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities, J. Differential Equations, 266 (2019), 5377-5396.  doi: 10.1016/j.jde.2018.10.030.  Google Scholar

[18]

D. GurbanP. Jebelean and C. Şerban, Nontrivial solutions for potential systems involving the mean curvature operator in Minkowski space, Adv. Nonlinear Stud., 17 (2017), 769-780.  doi: 10.1515/ans-2016-6025.  Google Scholar

[19]

Y.-H. Lee, Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain, Nonlinear Anal., 47 (2001), 3649-3660.  doi: 10.1016/S0362-546X(01)00485-0.  Google Scholar

[20]

R. MaT. Chen and H. Gao, On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator, Electron. J. Qual. Theory Differ. Equ., 98 (2016), 1-10.  doi: 10.14232/ejqtde.2016.1.98.  Google Scholar

[21]

J. E. Marsden and F. J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep., 66 (1980), 109-139.  doi: 10.1016/0370-1573(80)90154-4.  Google Scholar

[22]

R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., 65 (1979), 45-76.  doi: 10.1007/BF01940959.  Google Scholar

[23]

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

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