Skip to main content
SpringerLink
Log in

Multiplicity of positive radial solutions of singular Minkowski-curvature equations

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

This paper is about a singular Dirichlet problem involving the mean curvature operator in Minkowski space. We get the existence of triple and arbitrarily many positive radial solutions by using the Leggett-Williams fixed point theorem. Recent results in the literature are significantly improved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Am. Math. Soc. 137, 161–169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bereanu, C., Jebelean, P., Torres, P.J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264, 270–287 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bereanu, C., Jebelean, P., Torres, P.J.: Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J. Funct. Anal. 265, 644–659 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bereanu, C., Jebelean, P., Mawhin, J.: The Dirichlet problem with mean curvature operator in Minkowski space - a variational approach. Adv. Nonlinear Stud. 14, 315–326 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coelho, I., Corsato, C., Rivetti, S.: Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal. 44, 23–39 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Corsato, C., Obersnel, F., Omari, P., Rivetti, S.: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J. Math. Anal. Appl. 405, 227–239 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai, G.: Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space. Calc. Var. Partial Differ. Equ. 55, 1–17 (2016)

    Article  MathSciNet  Google Scholar 

  9. Gurban, D., Jebelean, P.: Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane-Emden type nonlinearities. J. Differ. Equ. 266, 5377–5396 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, S., Wang, H.: Convex solutions of boundary value problems arising from Monge-Ampère equations. Discret. Contin. Dyn. Syst. 16, 705–720 (2006)

    Article  MATH  Google Scholar 

  11. Leggett, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673–688 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ma, R., Gao, H., Lu, Y.: Global structure of radial positive solutions for a prescribed mean curvature problem in a ball. J. Funct. Anal. 270, 2430–2455 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ma, R.: Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space. Monatshefte für Math. 187, 315–325 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mawhin, J.: Radial solution of Neumann problem for periodic perturbations of the mean extrinsic curvature operator. Milan J. Math. 79, 95–112 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pei, M., Wang, L.: Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space. Appl. Math. Lett. 60, 50–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pei, M., Wang, L.: Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity. Proc. Am. Math. Soc. 145, 4423–4430 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to express our great thanks to the referee for his/her valuable suggestions. We also would like to show our thanks to Professor Jifeng Chu (Shanghai Normal University) for useful discussions.

Author information

Authors and Affiliations

  1. School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, 232001, Anhui, China

    Zaitao Liang, Lian Duan & Dandan Ren

  2. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, Hunan, China

    Lian Duan

Authors
  1. Zaitao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lian Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dandan Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zaitao Liang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Zaitao Liang was supported by the Natural Science Foundation of Anhui Province (1908085QA02) and the Key Program of Scientific Research Fund for Young Teachers of AUST (QN2018109); Lian Duan was supported by the National Natural Science Foundation of China (11701007) and China Postdoctoral Science Foundation (2018M640579); Dandan Ren was supported by the National Natural Science Foundation of China (11801008).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, Z., Duan, L. & Ren, D. Multiplicity of positive radial solutions of singular Minkowski-curvature equations. Arch. Math. 113, 415–422 (2019). https://doi.org/10.1007/s00013-019-01341-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-019-01341-6

Keywords

Mathematics Subject Classification

Search

Navigation