Skip to main content
SpringerLink
Log in

A Degenerate Kirchhoff-Type Inclusion Problem with Nonlocal Operator

  • Chapter
  • First Online:
Nonlinear Analysis and Global Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 167))

  • 853 Accesses

Abstract

The chapter focuses on a Kirchhoff-type elliptic inclusion problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points and for which the multivalued term is in the form of the generalized gradient of a locally Lipschitz function. The corresponding elliptic equation has been treated in (Liu et al., Existence of solutions to Kirchhoff-type problem with vanishing nonlocal term and fractional p-Laplacian). Multiple nontrivial solutions are obtained by applying the nonsmooth critical point theory combined with truncation techniques.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. K.C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)

    Article  MathSciNet  Google Scholar 

  5. F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. L. Gasiński, J.R. Santos Júnior, Multiplicity of positive solutions for an equation with degenerate nonlocal diffusion. Comput. Math. Appl. 78, 136–143 (2019)

    Article  MathSciNet  Google Scholar 

  9. Z.J. Guo, Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2015)

    Article  MathSciNet  Google Scholar 

  10. G. Kirchhoff, in Vorlesungen ueber Mathematische Physik, Mechanik. Lecture vol. 19 (Teubner, Leipzig, 1877)

    Google Scholar 

  11. Z.H. Liu, D. Motreanu, S. Zeng, Existence of solutions to Kirchhoff-type problem with vanishing nonlocal term and fractional p-Laplacian. Preprint

    Google Scholar 

  12. Z.H. Liu, J.G. Tan, Nonlocal elliptic hemivariational inequalities. Electron. J. Qual. Theory Differ. Equ. Paper 66 (2017), p. 7

    MATH  Google Scholar 

  13. D.F. Lü, S.J. Peng, Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type systems. J. Differ. Equ. 263, 8947–8978 (2017)

    Article  MathSciNet  Google Scholar 

  14. S. Migórski, S.D. Zeng, Mixed variational inequalities driven by fractional evolutionary equations. Acta Math. Sci. 39, 461–468 (2019)

    Article  MathSciNet  Google Scholar 

  15. S. Migórski, V.T. Nguyen, S.D. Zeng, Nonlocal elliptic variational-hemivariational inequalities. J. Integr. Equ. Appl. 32, 51–58 (2020)

    Article  MathSciNet  Google Scholar 

  16. O.H. Miyagaki, D. Motreanu, F.R. Pereira, Multiple solutions for a fractional elliptic problem with critical growth. J. Differ. Equ. 269, 5542–5572 (2020)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  18. D. Motreanu, P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Nonconvex Optimization and its Applications, vol. 29 (Kluwer Academic Publishers, Dordrecht, 1999)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. J.R. Santos Júnior, G. Siciliano, Positive solutions for Kirchhoff problems with vanishing nonlocal term. J. Differ. Equ. 265, 2034–2043 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. S.D. Zeng, Z.H. Liu, S. Migórski, A class of fractional differential hemivariational inequalities with application to contact problem. Z. Angew. Math. Phys. 69, 36 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Perpignan, Perpignan, France

    Dumitru Motreanu

Authors
  1. Dumitru Motreanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dumitru Motreanu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Motreanu, D. (2021). A Degenerate Kirchhoff-Type Inclusion Problem with Nonlocal Operator. In: Rassias, T.M., Pardalos, P.M. (eds) Nonlinear Analysis and Global Optimization. Springer Optimization and Its Applications, vol 167. Springer, Cham. https://doi.org/10.1007/978-3-030-61732-5_14

Download citation

Publish with us

Policies and ethics

Search

Navigation