Skip to main content
Log in

On a Fractional p-Laplacian Problem with Discontinuous Nonlinearities

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we are concerned by the study of a discontinuous elliptic problem involving a fractional p-Laplacian arising in differents context. Under suitable conditions, we provide the existence and multiplicity result via the nonsmooth critical point theory.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abdellaoui, B., Attar, A., Bentifour, R.: On the fractional p-Laplacian equations with weight and general datum. Adv. Nonlinear Anal. 8, 144–174 (2019)

    Article  MathSciNet  Google Scholar 

  2. Alexander, R.: A discontinuous nonlinear eigenvalue/Free boundary problem. Math. Methods Appl. Sci. 4(1), 131–142 (1982)

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti, A., Turner, R.E.L.: Some discontinuous variational problems. Differ. Integr. Equations 1, 341–348 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Ambrosetti, A., Badiale, M.: The dual variational principle and elliptic problems with discontinuous nonlinearities. J. Math. Anal. Appl. 140(2), 363–373 (1989)

    Article  MathSciNet  Google Scholar 

  5. Ambrosio, V.: A multiplicity result for a fractional p-Laplacian problem without growth conditions. Riv. Math. Univ. Parma 9(1), 53–71 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Ambrosio, V.: Nontrivial solutions for a fractional p-Laplacian problem via Rabier Theorem. Complex Var. Elliptic Equations 62(6), 838–847 (2017)

    Article  MathSciNet  Google Scholar 

  7. Ambrosio, V., Isernia, T.: Multiplicity and concentration results for some nonlinear Schrodinger equations with the fractional p-Laplacian. Discrete Contin. Dyn. Syst. 38(11), 5835–5881 (2018)

    Article  MathSciNet  Google Scholar 

  8. Arcoya, D., Calahorrano, M.: Some discontinuous problems with quasilinear operator. J. Math. Anal. Appl. 187, 1059–1072 (1994)

    Article  MathSciNet  Google Scholar 

  9. Bensid, S.: A discontinuous semilinear problem involving the fractional Laplacian. Nonlinear Stud. 24, 377–388 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Bensid, S.: Existence and multiplicity of solutions for fractional elliptic problems with discontinuous nonlinearities. Mediterr. J. Math. 15, 135 (2018)

    Article  MathSciNet  Google Scholar 

  11. Bensid, S., Bouguima, S.M.: A note on discontinuous problem with a free boundary. J. Egypt. Math. Soc. 19, 86–87 (2011)

    Article  MathSciNet  Google Scholar 

  12. Bensid, S., Bouguima, S.M.: Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions. Electron. J. Differ. Equations 56, 1–16 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Bensid, S., Bouguima, S.M.: On a free boundary problem. Nonlinear Anal. TMA 68, 2328–2348 (2008)

    Article  Google Scholar 

  14. Bensid, S., Kaid, Z.: Multiple stationary solutions of parabolic problem with discontinuous nonlinearities and their stability. Complex Var. Elliptic Equations, 1–20 (2020)

  15. Caffarelli, L.A.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equations Abel Symp. 7, 37–52 (2012)

    Article  Google Scholar 

  16. Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. 80, 102–129 (1981)

    Article  MathSciNet  Google Scholar 

  17. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 265, 247–262 (1975)

    Article  MathSciNet  Google Scholar 

  18. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  19. Ferrera, J.: An Introduction to Non-smooth Analysis. Elsevier Science, Amsterdam (2013)

    Google Scholar 

  20. Franzina, G., Palatucci, G.: Fractional p-eigenvalues. Riv. Mat. Univ. Parma, 795–826 (2013)

  21. Halidias, N.: Elliptic problems with discontinuities. J. Math. Anal. Appl. 276(1), 13–27 (2002)

    Article  MathSciNet  Google Scholar 

  22. Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory. Adv. Calc. Var. 9(2), 101–125 (2016)

    Article  MathSciNet  Google Scholar 

  23. Lehrer, R., Maia, L.A., Squassina, M.: On fractional \( p \)-Laplacian problems with weight. Differ. Integr. Equations 28(2014), 15–28 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Lieb, E.H., Loss, M.: Analysis. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  25. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial. Differ. Equations 49(1–2), 795–826 (2014)

    Article  MathSciNet  Google Scholar 

  26. Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis–Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differ. Equations 55(5), 105 (2016)

    Article  MathSciNet  Google Scholar 

  27. Mukherjee, T., Sreenadh, K.: On Dirichlet problem for fractional p-Laplacian with singular non-linearity. Adv. Nonlinear Anal. 8(1), 52–72 (2016)

    Article  MathSciNet  Google Scholar 

  28. Perera, K., Squassina, M., Yang, Y.: Bifurcation and multiplicity results for critical fractional p-Laplacian problems. Math. Nachr. 289(2–3), 332–342 (2016)

    Article  MathSciNet  Google Scholar 

  29. Piersanti, P., Pucci, P.: Existence theorems for fractional p-Laplacian problems. Anal. Appl. 15(5), 607–640 (2017)

    Article  MathSciNet  Google Scholar 

  30. Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)

    Article  MathSciNet  Google Scholar 

  31. Servadei, R., Valdinoci, E.: The Brezis Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)

    Article  MathSciNet  Google Scholar 

  32. Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)

    Article  MathSciNet  Google Scholar 

  33. Xiang, M., Zhang, B., Rǎdulescu, V.D.: Existence of solutions for perturbed fractional p-Laplacian equations. J. Differ. Equations 260(2), 1392–1413 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sabri Bensid.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Achour, H., Bensid, S. On a Fractional p-Laplacian Problem with Discontinuous Nonlinearities. Mediterr. J. Math. 18, 241 (2021). https://doi.org/10.1007/s00009-021-01898-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01898-z

Mathematics Subject Classification

Keywords

Navigation