DOI QR코드

DOI QR Code

S-SHAPED CONNECTED COMPONENT FOR A NONLINEAR DIRICHLET PROBLEM INVOLVING MEAN CURVATURE OPERATOR IN ONE-DIMENSION MINKOWSKI SPACE

  • Ma, Ruyun (Department of Mathematics Northwest Normal University) ;
  • Xu, Man (Department of Mathematics Northwest Normal University)
  • Received : 2018.01.04
  • Accepted : 2018.08.16
  • Published : 2018.11.30

Abstract

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$\{\(\frac{u^{\prime}}{\sqrt{1-u^{{\prime}2}}}\)^{\prime}+{\lambda}a(x)f(u)=0,\;x{\in}(0,1),\\u(0)=u(1)=0$$, where ${\lambda}$ is a positive parameter, $f{\in}C[0,{\infty})$, $a{\in}C[0,1]$. The proofs of main results are based upon the bifurcation techniques.

Acknowledgement

Supported by : NSFC

References

  1. L. J. Alias and B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc. 33 (2001), no. 4, 454-458. https://doi.org/10.1017/S0024609301008220
  2. R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), no. 1, 131-152. https://doi.org/10.1007/BF01211061
  3. 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), no. 1, 270-287. https://doi.org/10.1016/j.jfa.2012.10.010
  4. 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), no. 4, 644-659. https://doi.org/10.1016/j.jfa.2013.04.006
  5. C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular ${\phi}$-Laplacian, J. Differential Equations 243 (2007), no. 2, 536-557. https://doi.org/10.1016/j.jde.2007.05.014
  6. S. Cano-Casanova, J. Lopez-Gomez, and K. Takimoto, A quasilinear parabolic perturbation of the linear heat equation, J. Differential Equations 252 (2012), no. 1, 323-343. https://doi.org/10.1016/j.jde.2011.09.018
  7. S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407-419. https://doi.org/10.2307/1970963
  8. 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), no. 3, 621-638. https://doi.org/10.1515/ans-2012-0310
  9. 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), no. 1, 23-39.
  10. 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), no. 1, 113-134. https://doi.org/10.1515/gmj-2016-0078
  11. 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), no. 1, 227-239. https://doi.org/10.1016/j.jmaa.2013.04.003
  12. G. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 72, 17 pp.
  13. K.-C. Hung, S.-H. Wang, and C.-H. Yu, Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl. 392 (2012), no. 1, 40-54. https://doi.org/10.1016/j.jmaa.2012.02.036
  14. P. Korman, A global solution curve for a class of periodic problems, including the relativistic pendulum, Appl. Anal. 93 (2014), no. 1, 124-136. https://doi.org/10.1080/00036811.2012.762088
  15. H. J. Li and C. C. Yeh, Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1193-1204. https://doi.org/10.1017/S0308210500030468
  16. Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations 260 (2016), no. 12, 8358-8387. https://doi.org/10.1016/j.jde.2016.02.021
  17. R. Lopez, Stationary surfaces in Lorentz-Minkowski space, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 5, 1067-1096. https://doi.org/10.1017/S0308210507000273
  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), no. 2, 279-290. https://doi.org/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), no. 10, 4364-4376. https://doi.org/10.1016/j.na.2009.02.113
  20. R. Ma, T. Chen, and H. Gao, On positive solutions of the Dirichlet problem involving the extrinsic mean curvature operator, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 98, 10 pp.
  21. 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), no. 7, 2430-2455. https://doi.org/10.1016/j.jfa.2016.01.020
  22. R. Ma and Y. Lu, Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud. 15 (2015), no. 4, 789-803. https://doi.org/10.1515/ans-2015-0403
  23. H. Pan and R. Xing, Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions, J. Differential Equations 254 (2013), no. 3, 1464-1499. https://doi.org/10.1016/j.jde.2012.10.025
  24. P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487-513. https://doi.org/10.1016/0022-1236(71)90030-9
  25. T. Shibata, S-shaped bifurcation curves for nonlinear two-parameter problems, Nonlinear Anal. 95 (2014), 796-808. https://doi.org/10.1016/j.na.2013.10.015
  26. I. Sim and S. Tanaka, Three positive solutions for one-dimensional p-Laplacian problem with sign-changing weight, Appl. Math. Lett. 49 (2015), 42-50. https://doi.org/10.1016/j.aml.2015.04.007
  27. A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39-56. https://doi.org/10.1007/BF01404755
  28. W. Walter, Ordinary Differential Equations, translated from the sixth German (1996) edition by Russell Thompson, Graduate Texts in Mathematics, 182, Springer-Verlag, New York, 1998.
  29. S.-H. Wang and T.-S. Yeh, Exact multiplicity of solutions and S-shaped bifurcation curves for the p-Laplacian perturbed Gelfand problem in one space variable, J. Math. Anal. Appl. 342 (2008), no. 2, 1175-1191. https://doi.org/10.1016/j.jmaa.2007.12.026
  30. X. Xu, B. Qin and W. Li, S-shaped bifurcation curve for a nonlocal boundary value problem, J. Math. Anal. Appl. 450 (2017), no. 1, 48-62. https://doi.org/10.1016/j.jmaa.2016.12.073