SpringerLink
Log in

Multiple Solutions for a Kirchhoff-type Problem with Vanishing Nonlocal Term and Fractional p-Laplacian

  • Research Article
  • Published:
Frontiers of Mathematics Aims and scope Submit manuscript

Abstract

The goal of the paper is to investigate a Kirchhoff-type elliptic problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points. Multiple nontrivial solutions are obtained by applying a variational method combined with truncation techniques.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Autuori G., Fiscella A., Pucci P., Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal., 2015, 125: 699–714

    Article  MathSciNet  MATH  Google Scholar 

  2. Autuori G., Pucci P., Salvatori M.C., Global nonexistence for nonlinear Kirchhoff systems. Arch. Ration. Mech. Anal., 2010, 196: 489–516

    Article  MathSciNet  MATH  Google Scholar 

  3. Brézis H., Lieb E., A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc., 1983, 88: 486–490

    Article  MathSciNet  MATH  Google Scholar 

  4. Di Nezza E., Palatucci G., Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 2012, 136: 521–573

    Article  MathSciNet  MATH  Google Scholar 

  5. Figueiredo G.M., Santos Júnior J.R., Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity. J. Math. Phys., 2015, 56: 051506

    Article  MathSciNet  MATH  Google Scholar 

  6. Gasinski L., Santos Júnior J.R., Multiplicity of positive solutions for an equation with degenerate nonlocal diffusion. Comput. Math. Appl., 2019, 78: 136–143

    MathSciNet  MATH  Google Scholar 

  7. Gasiński L., Santos Júnior J.R., Nonexistence and multiplicity of positive solutions for an equation with degenerate nonlocal diffusion. Bull. Lond. Math. Soc., 2020, 52: 489–497

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo Z.J., Ground states for Kirchhoff equations without compact condition. J. Differential Equations, 2015, 259: 2884–2902

    Article  MathSciNet  MATH  Google Scholar 

  9. Kirchhoff G., Vorlesungen Über Mathematische Physik, Mechanik. Leipzig: Teubner, 1877

    MATH  Google Scholar 

  10. Liu Z.H., Tan J.G., Nonlocal elliptic hemivariational inequalities. Electron. J. Qual. Theory Differ. Equ., 2017, 2017: Paper No. 66: 7 pp.

  11. Lu D.F., Peng S.J., Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems. J. Differential Equations, 2017, 263: 8947–8978

    Article  MathSciNet  MATH  Google Scholar 

  12. Migónrski S., Nguyen V.T., Zeng S.D., Nonlocal elliptic variational-hemivariational inequalities. J. Integral Equations Appl., 2020, 32: 51–58

    MathSciNet  MATH  Google Scholar 

  13. Mignórski S., Zeng S.D., Mixed variational inequalities driven by fractional evolutionary equations. Acta Math. Sci. Ser. B Engl. Ed., 2019, 39: 461–468

    Article  MathSciNet  Google Scholar 

  14. Miyagaki O.H., Motreanu D., Pereira F.R., Multiple solutions for a fractional elliptic problem with critical growth. J. Differential Equations, 2020, 269: 5542–5572

    Article  MathSciNet  MATH  Google Scholar 

  15. Molica Bisci G., Rădulescu V.D., Servadei R., Variational Methods for Nonlocal Fractional Problems. Cambridge: Cambridge University Press, 2016

    Book  MATH  Google Scholar 

  16. Pan N., Zhang B.L., Cao J., Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian. Nonlinear Anal. Real World Appl., 2017, 37: 56–70

    Article  MathSciNet  MATH  Google Scholar 

  17. Perera K., Zhang Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differential Equations, 2006, 221: 246–255

    Article  MathSciNet  MATH  Google Scholar 

  18. Ricceri B., Energy functionals of Kirchhoff-type problems having multiple global minima. Nonlinear Anal., 2015, 115: 130–136

    Article  MathSciNet  MATH  Google Scholar 

  19. Santos Júnior J.R., Siciliano G., Positive solutions for Kirchhoff problems with vanishing nonlocal term. J. Differential Equations, 2018, 265: 2034–2043

    Article  MathSciNet  MATH  Google Scholar 

  20. **ang M.Q., Rădulescu V.D., Zhang B.L., Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity, 2018, 31: 3228–3250

    Article  MathSciNet  MATH  Google Scholar 

  21. **ang M.Q., Zhang B.L., Ferrara M., Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian. J. Math. Anal. Appl., 2015, 424: 1021–1041

    Article  MathSciNet  MATH  Google Scholar 

  22. Zeng S.D., Liu Z.H., Migórski S., A class of fractional differential hemivariational inequalities with application to contact problem. Z. Angew. Math. Phys., 2018, 69: Paper No. 36, 23 pp.

Download references

Acknowledgements

This project has received funding from the Natural Science Foundation of Guangxi (Nos. 2021GXNSFFA196004, GKAD23026237), the NNSF of China (Nos. 12001478, 12071413, 12371312), the China Postdoctoral Science Foundation Funded Project (No. 2022M721560), and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska–Curie grant agreement (No. 823731) CONMECH. It is also supported by the Ministry of Science and Higher Education of Republic of Poland (Nos. 4004/GGPJII/H2020/2018/0, 440328/PnH2/2019), and the project cooperation between Guangxi Normal University and Yulin Normal University.

Author information

Authors and Affiliations

  1. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, China

    Zhenhai Liu

  2. Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning, 530006, China

    Zhenhai Liu

  3. Département de Mathématiques, Université de Perpignan, 66860, Perpignan, France

    Dumitru Motreanu

  4. Department of Mathematics, Nan**g University, Nan**g, 210093, China

    Shengda Zeng

  5. Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348, Krakow, Poland

    Shengda Zeng

Authors
  1. Zhenhai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dumitru Motreanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shengda Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengda Zeng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Motreanu, D. & Zeng, S. Multiple Solutions for a Kirchhoff-type Problem with Vanishing Nonlocal Term and Fractional p-Laplacian. Front. Math 18, 1067–1082 (2023). https://doi.org/10.1007/s11464-021-0019-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-021-0019-5

Keywords

MSC2020

Search

Navigation