Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Multiplicity results for a nonlocal fractional problem

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper presents new multiplicity and existence results for a class of nonlocal fractional problems. Based on critical point theory and the mountain pass theorem, we derive conditions ensuring the multiplicity and existence of weak solution for the fractional problem in a suitable space of functions.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /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 excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  • Ambrosetti A, Rabinowitz PH (1973) Dual variational methods in critical point theory and applications. J Funct Anal 14(4):349–381

    Article  MathSciNet  Google Scholar 

  • Ayoujil A, El Amrouss A (2009) On the spectrum of a fourth order elliptic equation with variable exponent. Nonlinear Anal Theory Methods Appl 71(10):4916–4926

    Article  MathSciNet  Google Scholar 

  • Azroul E, Boumazourh A (2020) Three solution for a fractional \((p (x,.), q (x,.))\)-Kirchhoff type elliptic system. Nonlinear Funct Anal 2020:Article ID 40

  • Azroul E, Benkirane A, Shimi M, Srati M (2020) Three solutions for fractional \(p (x,)\)-Laplacian Dirichlet problems with weight. J Nonlinear Funct Anal 2020:1–18

    MATH  Google Scholar 

  • Azroul E, Benkirane A, Boumazourh A, Shimi M (2021) Existence results for fractional p (x,)-Laplacian problem via the Nehari manifold approach. Appl Math Optim 84(2):1527–1547

    Article  MathSciNet  Google Scholar 

  • Bahrouni A, Rădulescu VD (2018) On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin Dyn Syst 11(3):379

    MathSciNet  MATH  Google Scholar 

  • Bonanno G (2012) A critical point theorem via the Ekeland variational principle. Nonlinear Anal Theory Methods Appl 75(5):2992–3007

    Article  MathSciNet  Google Scholar 

  • Boureanu MM, Rădulescu V, Repovš D (2016) On a \(p (\cdot )\)-biharmonic problem with no-flux boundary condition. Comput Math Appl 72(9):2505–2515

    Article  MathSciNet  Google Scholar 

  • Caffarelli L (2012) Non-local diffusions, drifts and games. In: Nonlinear partial differential equations, Springer, pp 37–52

  • Chung NT, Naghizadeh Z (2021) Multiplicity of solutions for a class of fourth-order elliptic equations of \(p(x)\)-Kirchhoff type. Math Slovaca 71(6):1441–1458

    Article  MathSciNet  Google Scholar 

  • Del Pezzo L, Rossi JD (2017) Traces for fractional Sobolev spaces with variable exponents. Adv Oper Theory 2(4):435–446

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Diening L, Harjulehto P, Hästö P, Ruzicka M (2011) Lebesgue and Sobolev spaces with variable exponents, vol 2017. Springer, New York

  • Fan X, Zhao D (2001) On the spaces \({L}^{p (x)}({\Omega })\) and \({W}^{m, p (x)}({\Omega })\). J Math Anal Appl 263(2):424–446

    Article  MathSciNet  Google Scholar 

  • Ferrara M, Khademloo S, Heidarkhani S (2014) Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl Math Comput 234:316–325

    Article  MathSciNet  Google Scholar 

  • Hamdani MK, Zuo J, Chung NT (2020) Repovš D (2020) Multiplicity of solutions for a class of fractional \(p (x,) \)-Kirchhoff-type problems without the Ambrosetti-Rabinowitz condition. Bound Value Problem 1:1–16

    Google Scholar 

  • Jarohs S (2018) Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings. Adv Nonlinear Stud 18(4):691–704

    Article  MathSciNet  Google Scholar 

  • Kaufmann U, Rossi JD, Vidal RE (2017) Fractional Sobolev spaces with variable exponents and fractional \(p (x)\)-Laplacians

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

    Article  MathSciNet  Google Scholar 

  • Radulescu VD, Repovs DD (2015) Partial differential equations with variable exponents: variational methods and qualitative analysis, vol 9. CRC Press, Boca Raton

    Book  Google Scholar 

  • Zang A, Fu Y (2008) Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Anal Theory Methods Appl 69(10):3629–3636

    Article  MathSciNet  Google Scholar 

  • Zeidler E (2013) Nonlinear functional analysis and its applications: II/B: nonlinear monotone operators. Springer, New York

Download references

Acknowledgements

The authors would like to thank Associate Editor and three anonymous reviewers for their comments on the manuscript which helped very much in improving and presenting the original version of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Z. Naghizadeh.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Communicated by Agnieszka Malinowska.

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

Naghizadeh, Z., Nikan, O. & Lopes, A.M. Multiplicity results for a nonlocal fractional problem. Comp. Appl. Math. 41, 239 (2022). https://doi.org/10.1007/s40314-022-01931-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01931-1

Keywords

Mathematics Subject Classification

Navigation