Skip to main content
SpringerLink
Log in

Infinitely Solutions for a Fractional \(p(\cdot ,\cdot )\)-Kirchhoff Type Problem

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

The purpose of this paper is to study the existence and infinitely many solutions for a some type of nonlocal \(p(\cdot ,\cdot )\)-Kirchhoff problems with Dirichlet boundary conditions. Our proofs are based on variational methods, the Mountain Pass Lemma, and genus theory.

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

Data Availability

I declare the data availability in the manuscript.

References

  1. Ali, K.B., Hsini, M., Kefi, K., Chung, N.T.: On a nonlocal fractional \(p(.,.)\)-Laplacian problem with competing nonlinearities. Complex Anal. Oper. Theory 13(3), 1377–1399 (2019)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies Advanced Mathematics, vol. 14 (2007)

  3. Antontsev, S.N., Shmarev, S.I.: A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)

    Article  MathSciNet  Google Scholar 

  4. Castro, A.: Metodos variacionales y analisis functional no linear. In: X Coloquio Colombiano de Matematicas (1980)

  5. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  6. Clarke, D.C.: A variant of the Lusternik–Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972)

    Article  MathSciNet  Google Scholar 

  7. Coelho, V., Ribeiro, J., Salgado, L.: On Schauder basis in normed spaces, pp. 1–9 (2018) arXiv:1806.07943v1 [math.FA]

  8. Correa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of p-Kirchhoff-type via variational methods. Bull. Aust. Math. Soc. 74, 263–277 (2006)

    Article  MathSciNet  Google Scholar 

  9. Correa, F.J.S.A., Figueiredo, G.M.: On a p-Kirchhoff equation via Krasnoselskii’s genus. Appl. Math. Lett. 22, 819–822 (2009)

    Article  MathSciNet  Google Scholar 

  10. Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}\) and \(W^{m, p(x)}\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  Google Scholar 

  11. Fiscella, A.: A fractional Kirchhoff problem involving a singular term and a critical nonlinearity. Adv. Nonlinear Anal. 8, 645–660 (2019)

    Article  MathSciNet  Google Scholar 

  12. Ghanmi, A., Kratou, M., Saoudi, K., Repovs, D.D.: Nonlocal \(p\)-Kirchhoff equations with singular and critical nonlinearity terms. Asympt. Anal. 131(1), 125–143 (2023)

    MathSciNet  Google Scholar 

  13. Hamdani, M.K., Zuo, J., Chung, N.T., Repovs̆, D.D.: Multiplicity of solutions for a class of fractional \(p(x,\cdot )\)-Kirchhoff type problems without the Ambrosetti–Rabinowitz condition. Bound. Value Probl. https://doi.org/10.1186/s13661-020-01447-9

  14. Kaufmann, U., Rossi, J.D., Vidal, R.: Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electron. J. Qual. Theory Differ. Equ. 76, 1–10 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kavian, O.: Introduction a la theorie des points critiques et applications aux problemes elliptiques. Springer, Berlin (1993)

    Google Scholar 

  16. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    Google Scholar 

  17. Kratou, M.: Kirchhoff systems involving fractional \(p\)-Laplacian and singular non-linearity. Electron. J. Differ. Equ. 2022(77), 1–15 (2022)

    MathSciNet  Google Scholar 

  18. Kratou, M.: Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann–Liouville fractional derivative. Filomat 33(7), 2073–2088 (2019)

    Article  MathSciNet  Google Scholar 

  19. Krasnoselskii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York (1964)

    Google Scholar 

  20. Lei, C.Y., Liao, J.F., Tang, C.L.: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J. Math. Anal. Appl. 421, 521–538 (2015)

    Article  MathSciNet  Google Scholar 

  21. Liao, J.F., Ke, X.F., Lei, C.Y., Tang, C.L.: A uniqueness result for Kirchhoff type problems with singularity. Appl. Math. Lett. 59, 24–30 (2016)

    Article  MathSciNet  Google Scholar 

  22. Liao, J.F., Zhang, P., Liu, J., Tang, C.L.: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity. J. Math. Anal. Appl. 430, 1124–1148 (2015)

    Article  MathSciNet  Google Scholar 

  23. Liu, X., Sun, Y.: Multiple positive solutions for Kirchhoff type of problems with singularity. Commun. Pure Appl. Anal. 12, 721–733 (2013)

    MathSciNet  Google Scholar 

  24. Liu, R.Q., Tang, C.L., Liao, J.F., Wu, X.P.: Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Commun. Pure Appl. Anal. 15, 1841–1856 (2016)

    Article  MathSciNet  Google Scholar 

  25. Mokhtari, A., Moussaoui, T., O’Regan, D.: Existence and multiplicity of solutions for a \(p(x)\)-Kirchhoff type problem via variational techniques. Archivum Mathematicum 51(3), 163–173 (2015)

    Article  MathSciNet  Google Scholar 

  26. Mokhtari, A., Saoudi, K., Zuo, J.: Critical p(x)-Kirchhoff problems involving variable singular exponent. Bull. Iran. Math. Soc. 48, 2917–2942 (2022)

    Article  MathSciNet  Google Scholar 

  27. Molica Bisci, G., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, Encyclopedia Mathematics Applications, vol. 162. Cambridge University Press, Cambridge (2016)

  28. Peral, I.: Multiplicity of Solutions for the p-Laplacian. Second School of Nonlinear Functional Analysis and Applications to Differential Equations. ICTP, Trieste (1997)

  29. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: Conference Board of the Mathematical Sciences, by the American Mathematical Society Providence, Rhode Island (1984)

  30. Ruẑička, M.: Electro-rheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

    Book  Google Scholar 

  31. Saoudi, K.: A fractional Kirchhof system with singular nonlinearities. Anal. Math. Phys. 9, 1463–1480 (2019)

    Article  MathSciNet  Google Scholar 

  32. Singer, I.: Bases in Banach Spaces I. Springer, Berlin (1970)

    Book  Google Scholar 

  33. Singer, I.: Bases in Banach Spaces II. Springer, Berlin (1981)

    Book  Google Scholar 

  34. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710 (1986); English transl., Math. USSR-Izv. 29(1), 33–66 (1987)

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their carefully reading this paper and their useful comments.

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of mathematics and informatics, University of M’sila, M’sila, Algeria

    Abdelhak Mokhtari

  2. Laboratory of Fixed Point Theory and Applications, Department of Mathematics, E.N.S. Kouba, Algiers, Algeria

    Abdelhak Mokhtari

  3. Basic and applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Kingdom of Saudi Arabia

    Mouna Kratou & Kamel Saoudi

Authors
  1. Abdelhak Mokhtari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mouna Kratou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kamel Saoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors reviewed the manuscript.

Corresponding author

Correspondence to Kamel Saoudi.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by Jussi Behrndt.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mokhtari, A., Kratou, M. & Saoudi, K. Infinitely Solutions for a Fractional \(p(\cdot ,\cdot )\)-Kirchhoff Type Problem. Complex Anal. Oper. Theory 18, 70 (2024). https://doi.org/10.1007/s11785-024-01519-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-024-01519-z

Search

Navigation