Skip to main content
SpringerLink
Log in

A Hardy–Littlewood–Sobolev-Type Inequality for Variable Exponents and Applications to Quasilinear Choquard Equations Involving Variable Exponent

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

Abstract

In this work, we have proved a Hardy–Littlewood–Sobolev inequality for variable exponents. After that, we use this inequality together with the variational method to establish the existence of solution for a class of Choquard equations involving the p(x)-Laplacian operator.

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

References

  1. Acerbi, E., Mingione, G.: Regularity results for tationary electrorheological fluids. Arch. Rational Mech. Anal. 164, 213–259 (2002)

    Article  MathSciNet  Google Scholar 

  2. Acerbi, E., Mingione, G.: Regularity results for electrorheological fluids: stationary case. C.R. Math. Acad. Sci. Paris 334, 817–822 (2002)

    Article  MathSciNet  Google Scholar 

  3. Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)

    Article  MathSciNet  Google Scholar 

  4. Almeida, A., Samko, S.: Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Funct. Spaces Appl. 4(2), 113–144 (2006)

    Article  MathSciNet  Google Scholar 

  5. Alves, C.O.: Existence of solutions for a degenerate \(p(x)\)-Laplacian equation in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 345, 731–742 (2008)

    Article  MathSciNet  Google Scholar 

  6. Alves, C.O.: Existence of radial solutions for a class of \(p(x)\)-Laplacian equations with critical growth. Diff. Integral Equ. 23, 113–123 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Alves, C.O., Barreiro, J.L.P.: Existence and multiplicity of solutions for a \(p(x)\)-Laplacian equation with critical growth. J. Math. Anal. Appl. 403, 143–154 (2013)

    Article  MathSciNet  Google Scholar 

  8. Alves, C.O., Ferreira, M.C.: Nonlinear perturbations of a \(p(x)\)-Laplacian equation with critical growth in \(\mathbb{R}^{N}\). Math. Nachr. 287(8–9), 849–868 (2014)

    Article  MathSciNet  Google Scholar 

  9. Alves, C.O., Ferreira, M.C.: Existence of solutions for a class of \(p(x)\)-Laplacian equations involving a concave-convex nonlinearity with critical growth in \(\mathbb{R}^{N}\). Topol. Methods Nonlinear Anal. 45(2), 399–422 (2014)

    Article  Google Scholar 

  10. Alves, C.O., Moussaoui, A.: Positive solutions for a class of quasilinear singular elliptic systems, Submitted

  11. Alves, C.O., Souto, M.A.S.: Existence of solutions for a class of problems in \(\mathbb{R}^{N}\) involving \(p(x)\)-Laplacian. Prog. Nonlinear Diff. Equ. Appl. 66, 17–32 (2005)

    Google Scholar 

  12. Alves, C., Liu, S.: On superlinear \(p(x)-\)Laplacian equations in \(\mathbb{R}^N\). Nonlinear Anal. 73(8), 2566–2579 (2010)

    Article  MathSciNet  Google Scholar 

  13. Alves, C.O., Yang, M.: Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method. Proc. R. Soc. Edinb. Sect. A 146, 23–58 (2016)

    Article  Google Scholar 

  14. Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257, 4133–4164 (2014)

    Article  MathSciNet  Google Scholar 

  15. Antontsev, S.N., Rodrigues, J.F.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, 19–36 (2006)

    Article  MathSciNet  Google Scholar 

  16. Chabrowski, J.: Variational methods for potential operator equations with applications to nonlinear elliptic equations. Walter de Gruyter, Berlin-New York (1997)

    Book  Google Scholar 

  17. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Cingolani, S., Secchi, S., Squassina, M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinb. Sect. A 140, 973–1009 (2010)

    Article  Google Scholar 

  20. Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev spaces with variable exponents, Lectures Notes in Math, vol. 2017. Springer-Verlag, Heidelberg (2011)

    Book  Google Scholar 

  21. Fan, X.: Global \(C^{1,\alpha }\) regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 235, 397–417 (2007)

    Article  Google Scholar 

  22. Fan, X.L.: On the sub-supersolution method for \(p(x)\) -Laplacian equations. J. Math. Anal. Appl. 330, 665–682 (2007)

    Article  MathSciNet  Google Scholar 

  23. Fan, X.L.: \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\) with periodic data and nonperiodic perturbations. J. Math. Anal. Appl. 341, 103–119 (2008)

    Article  MathSciNet  Google Scholar 

  24. Fan, X., Zhao, D.: A class of De Giorgi type and H ölder continuity. Nonlinear Anal. 36, 295–318 (1999)

    Article  MathSciNet  Google Scholar 

  25. Fan, X.L., Zhao, D.: On the Spaces \(L^{p(x)}\big (\Omega \big )\) and \(W^{1, p(x)}\big (\Omega \big )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  Google Scholar 

  26. Fan, X.L., Zhao, D.: Nodal solutions of \(p(x)\)-Laplacian equations. Nonlinear Anal. 67, 2859–2868 (2007)

    Article  MathSciNet  Google Scholar 

  27. Fan, X.L., Shen, J.S., Zhao, D.: Sobolev embedding theorems for spaces \(W^{k, p(x)}\big (\Omega \big )\). J. Math. Anal. Appl. 262, 749–760 (2001)

    Article  MathSciNet  Google Scholar 

  28. Fan, X., Zhang, Q., Zhao, D.: Eigenvalues of \(p(x)-\) Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306–317 (2005)

    Article  MathSciNet  Google Scholar 

  29. Fan, X.L., Zhao, Y.Z., Zhang, Q.H.: A strong maximum principle for p(x)-Laplace equations Chinese. J. Contemp. Math. 24(3), 277–282 (2003)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Fernández Bonder, J., Saintier, N., Silva, A.: On the Sobolev embedding theorem for variable exponent spaces in the critical range. J. Differ. Equ. 253, 1604–1620 (2012)

    Article  MathSciNet  Google Scholar 

  32. Fernández Bonder, J., Saintier, N., Silva, A.: On the Sobolev trace theorem for variable exponent spaces in the critical range. Ann. Mat. Pura Appl. 6, 1607–1628 (2014)

    Article  MathSciNet  Google Scholar 

  33. Fu, Y., Zhang, X.: Multiple solutions for a class of \(p(x)\)-Laplacian equations in involving the critical exponent. Proc. R. Soc. Edinb. Sect. A 466, 1667–1686 (2010)

    Article  MathSciNet  Google Scholar 

  34. Hajibayov, M.G., Samko, S.: Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces. Math. Nachr. 284(1), 53–66 (2011)

    Article  MathSciNet  Google Scholar 

  35. Gao, F., Yang, M.: On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, arXiv:1604.00826v4

  36. Kokilashvili, V., Meskhi, A.: Boundedness of maximal and singular operators in Morrey spaces with variable exponent. Armen. J. Math. 1(1), 18–28 (2008)

    MathSciNet  MATH  Google Scholar 

  37. Kokilashvili, V., Samko, S.: On Sobolev theorem for Riesz-type potentials in Lebesgue spaces with variable exponent. Z. Anal. Anwend. 22(4), 899–910 (2003)

    Article  MathSciNet  Google Scholar 

  38. Kokilashvili, V., Samko, S.: Weighted boundedness of the maximal, singular and potential operators in variable exponent spaces in Analytic Methods of Analysis and Differential Equations, Kilbas, A.A., Rogosin, S.V., eds., Cambridge Scientific Publishers, Cambridge, pp. 139-164 (2008)

  39. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57, 93–105, (1976/77)

    Article  MathSciNet  Google Scholar 

  40. Lieb, E., Loss, M.: Analysis. Gradute Studies in Mathematics, AMS, Providence, Rhode island (2001)

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

    Article  MathSciNet  Google Scholar 

  42. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)

    Article  MathSciNet  Google Scholar 

  43. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math, vol. 1034. Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  44. Mihăilescu, M., Rădulescu, V.: On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proc. Am. Math. Soc. 135(9), 2929–2937 (2007)

    Article  MathSciNet  Google Scholar 

  45. Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Class. Quantum Gravity 15, 2733–2742 (1998)

    Article  Google Scholar 

  46. Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

    Article  MathSciNet  Google Scholar 

  47. Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Part. Differ. Equ. 52, 199–235 (2015)

    Article  MathSciNet  Google Scholar 

  48. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent. Commun. Contemp. Math., 17, 1550005, 12 pp. (2015)

    Article  MathSciNet  Google Scholar 

  49. Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  50. Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)

    Article  MathSciNet  Google Scholar 

  51. Rădulescu, V., Repovs̆, D.: Partial differential equations with variable exponents. Variational methods and qualitative analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)

    Book  Google Scholar 

  52. Ruzicka, M.: Electrorheological fluids: Modeling and mathematical theory. Lecture Notes in Math., vol. 1748. Springer-Verlag, Berlin (2000)

    Book  Google Scholar 

  53. Samko, S.G.: Convolution type operators in \(L^{p(x)}\). Integral Transforms Spec. Funct. 7(1–2), 123–144 (1998)

    Article  MathSciNet  Google Scholar 

  54. Samko, S.G.: Convolution and potential type operators in \(L^{p(x)}(\mathbb{R}^n)\). Integral Transforms Spec. Funct. 7(3–4), 261–284 (1998)

    Article  MathSciNet  Google Scholar 

  55. Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct. 16(5–6), 461–482 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was done while the second author was visiting the Federal University of Campina Grande. He thanks the hospitality of professor Claudianor Alves and of the other members of the department. The authors warmly thank the anonymous referee for his/her useful and nice comments on the paper.

Author information

Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP: 58429-900, Brazil

    Claudianor O. Alves & Leandro S. Tavares

Authors
  1. Claudianor O. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leandro S. Tavares
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leandro S. Tavares.

Additional information

Publisher's Note

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

C.O. Alves was partially supported by CNPq/Brazil 301807/2013-2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s00009-019-1316-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-019-1316-z

Mathematics Subject Classification

Keywords

Search

Navigation