Skip to main content
Log in

Strauss and Lions Type Theorems for the Fractional Sobolev Spaces with Variable Exponent and Applications to Nonlocal Kirchhoff–Choquard Problem

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

Abstract

This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent \(W^{s,p(.),{\tilde{p}}(.,.)} (\Omega )\), when p and \({\tilde{p}}\) satisfy some conditions. As application, we study the existence of solutions for a class of Kirchhoff–Choquard problem in \({\mathbb {R}}^N\).

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

  1. Alves, C.O., Tavares, L.S.: A Hardy–Littlewood–Sobolev type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent. Mediterr. J. Math. 16, 55 (2019)

    Article  MathSciNet  Google Scholar 

  2. Alves, C., R��dulescu, V.D., Tavares, L.: Generalized Choquard equations driven by nonhomogeneous operators. Mediterr. J. Math. 16(1), 24 (2019) (Paper No. 20)

  3. Alves, C.O., Figueiredo, G.M., Santos, J.A.: Strauss and Lions type results for a class of Orlicz–Sobolev spaces and applications. Topol. Methods Nonlinear Anal. 44, 435–456 (2014)

    Article  MathSciNet  Google Scholar 

  4. Alves, C.O., Molica Bisci, G.: A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.123490

    Article  Google Scholar 

  5. Ali, K.B., Hsini, M., Kefi, K., Chung, N.T.: On a nonlocal fractional \(p(.,.)\)-Laplacian problem with competing nonlinearities. Complex Anal. Oper. Theory (2019). https://doi.org/10.1007/s11785-018-00885-9

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

    Article  MathSciNet  Google Scholar 

  7. Azroul, E., Benkirane, A., Shimi, M.: Aigenvalue problems involving the fractional \(p(x)\)-Laplacian operator. Adv. Oper. Theory. 4, 539–555 (2019)

    Article  MathSciNet  Google Scholar 

  8. Azroul, E., Benkirane, A., Shimi, M.: An introduction to generalized fractional sobolev space with variable exponent (1901). arXiv:1901.05687v1

  9. Bahrouni, A.: Comparison and sub-supersolution principles for the fractional \(p(x)-\)Laplacian. J. Math. Anal. Appl. 458, 1363–1372 (2018)

    Article  MathSciNet  Google Scholar 

  10. Bahrouni, A., Radulescu, V.: On a new fractional Sobolev space and application to nonlocal variational problems with variable exponent. Discrete Contin. Dyn. Syst. Ser. S 11, 379–389 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Biswas, R., Tiwari, S.: Variable order nonlocal Choquard problem with variable exponents (2020). arxiv:1907.02837

  12. Caponi, M., Pucci, P.: Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations. Ann. Mat. Pura Appl. 195, 2099–2129 (2016)

    Article  MathSciNet  Google Scholar 

  13. Cassani, D., Zhang, J.: Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth. Adv. Nonlinear Anal. 8(1), 1184–1212 (2019)

    Article  MathSciNet  Google Scholar 

  14. Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)

  15. Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of \({\mathbb{R}}^n\), Lecture Notes, Scuola Normale Superiore di Pisa, 15. Edizioni della Normale, Pisa (2017)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Fan, X., Zhao, Y., Zhao, D.: Compact embedding theorems with symmetry of Strauss–Lions type for the space \(W^{1, p}(\Omega )\). J. Math. Anal. Appl. 255, 333–348 (2001)

    Article  MathSciNet  Google Scholar 

  19. Ho, K., Kim, Y.-H.: A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional \(p(x)\)-Laplacian. Nonlinear Anal. 188, 179–201 (2019)

    Article  MathSciNet  Google Scholar 

  20. Ho, K., Sim, I.: Properties of eigenvalues and some regularities on fractional \(p\)-Laplacian with singular weights (2019). arXiv:1809.07020v1

  21. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{m, p(x)}\). Czechoslov. Math. J. 41, 592–618 (1991)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Lions, P.L.: Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)

    Article  Google Scholar 

  24. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/1977)

  25. Lions, P.L.: The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980)

    Article  MathSciNet  Google Scholar 

  26. Bisci, G.M., Rădulescu, V., Servadei, R.: Variational methods for nonlocal fractional problems, encyclopedia of mathematics and its applications, vol. 162. Cambridge University Press, Cambridge (2016)

  27. de Morais Filho, D.C., Souto, M. A.S., Marcos do, J.M.: A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems. Proyecciones 19, 1–17 (2000)

  28. Moroz, V., Schaftingen, J.V.: Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

    Article  MathSciNet  Google Scholar 

  29. Menzala, G.P.: On regular solutions of a nonlinear equation of Choquard’s type. Proc. R. Soc. Edinb. Sect. 86(3–4), 291–301 (1980)

    Article  MathSciNet  Google Scholar 

  30. de Nápoli, P., Drelichman, I.: Elementary proofs of embedding theorems for potential spaces of radial functions. Methods of Fourier analysis and approximation theory. Appl. Numer. Harmon. Anal., https://doi.org/10.1007/978-3-319-27466-9-8

  31. Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)

    Article  MathSciNet  Google Scholar 

  32. Pan, H.L., Liu, J., Tang, C.L.: Existence of a positive solution for a class of Choquard equation with upper critical exponent. Differ. Equations Dya. Syst. (2018). https://doi.org/10.1007/s12591-018-0437-3

  33. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)

    Article  MathSciNet  Google Scholar 

  34. Wang, L.: Multiple solutions for nonhomogeneous Choquard equations. Electron J. Differ. Equations 172, 1–27 (2018)

    MathSciNet  Google Scholar 

  35. Xiang, M., Zhang, B, Yang, Di.: Multiplicity results for variable-order fractional Laplacian equations with variable growth. Nonlinear Anal. 178, 190–204 (2019)

  36. Xiang, M., Rădulescu, V.D., Zhang, B.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional \(p\)-Laplacian and critical exponent. Adv. Nonlin. Anal. 9, 690–709 (2020)

    Article  Google Scholar 

  37. Xiang, M., Rădulescu, V.D., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equations 58(2), 57 (2019)

  38. Yuan, W., Sickel, W., Yang, D.: The radial lemma of Strauss in the context of Morrey spaces. Ann. Acad. Sci. Fenn Math. 39, 417–422 (2014)

    Article  MathSciNet  Google Scholar 

  39. Zhang, C., Zhang, X.: Renormalized solutions for the fractional \(p(x)\)-Laplacian equation with \(L^1\) data (2020). arXiv:1708.04481v1

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sabri Bahrouni.

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

Bahrouni, S., Ounaies, H. Strauss and Lions Type Theorems for the Fractional Sobolev Spaces with Variable Exponent and Applications to Nonlocal Kirchhoff–Choquard Problem. Mediterr. J. Math. 18, 46 (2021). https://doi.org/10.1007/s00009-020-01661-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-01661-w

Keywords

Mathematics Subject Classification

Navigation