\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Petru Jebelean

    * Corresponding author: Petru Jebelean 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35J66, 34B16; Secondary: 34B18.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [8] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Pure Math., 15 (1970), 223-230. 
    [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.
    [10] Y. Choquet-BruhatGeneral Relativity and the Einstein Equations, Oxford University Press, 2009. 
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [23] A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39-56.  doi: 10.1007/BF01404755.
  • 加载中
SHARE

Article Metrics

HTML views(171) PDF downloads(293) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return