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Generalized Choquard Equations Driven by Nonhomogeneous Operators

  • Claudianor O. Alves
  • Vicenţiu D. RădulescuEmail author
  • Leandro S. Tavares
Open Access
Article
  • 53 Downloads

Abstract

In this work we prove the existence of solutions for a class of generalized Choquard equations involving the \(\Delta _\Phi \)-Laplacian operator. Our arguments are essentially based on variational methods. One of the main difficulties in this approach is to use the Hardy–Littlewood–Sobolev inequality for nonlinearities involving N-functions. The methods developed in this paper can be extended to wide classes of nonlinear problems driven by nonhomogeneous operators.

Keywords

Choquard equation variational methods nonlinear elliptic equation Hardy–Littlewood–Sobolev inequality 

Mathematics Subject Classification

35A15 35J62 35J60 

Notes

Acknowledgements

This work started when Leandro S. Tavares was visiting the Federal University of Campina Grande. He thanks the hospitality of Professor Claudianor Alves and of the other members of the department. V. D. Rădulescu was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. He also acknowledges the support through the Project MTM2017-85449-P of the DGISPI (Spain). C. O. Alves was partially supported by CNPq/Brazil 301807/2013-2.

References

  1. 1.
    Fröhlich, H.: Theory of electrical breakdown in ionic crystal. Proc. R. Soc. Ser. A 160(901), 230–241 (1937)Google Scholar
  2. 2.
    Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie, Berlin (1954)zbMATHGoogle Scholar
  3. 3.
    Lieb, E. H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/77)Google Scholar
  4. 4.
    Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Class. Quant. Grav. 15, 2733–2742 (1998)CrossRefGoogle Scholar
  5. 5.
    Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alves, C.O., Figueiredo, G.M., Yang, M.: Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity. Adv. Nonlinear Anal. 5(4), 331–345 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Alves, C.O., Yang, M.: Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method. Proc. R. Soc. Edinburgh Sect. A 146, 23–58 (2016)CrossRefGoogle Scholar
  8. 8.
    Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257, 4133–4164 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cingolani, S., Secchi, S., Squassina, M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinburgh Sect. A 140, 973–1009 (2010)CrossRefGoogle Scholar
  10. 10.
    Gao, F., Yang, M.: On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. Sci. China Math. 61, 1219–1242 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lions, P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Moroz, V., van Schaftingen, J.: Existence of ground states for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)CrossRefGoogle Scholar
  14. 14.
    Moroz, V., van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52, 199–235 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Moroz, V., van Schaftingen, J.: Ground states of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun. Contemp. Math. 17, 12 (2015)CrossRefGoogle Scholar
  16. 16.
    Moroz, V., van Schaftingen, J.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19(1), 773–813 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    van Schaftingen, J., Xia, J.: Standing waves with a critical frequency for nonlinear Choquard equations. Nonlinear Anal. 161, 87–107 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang, T.: Existence of positive ground-state solution for Choquard-type equations. Mediterr. J. Math. 14(1), 15 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics. Amer. Math. Soc, Providence (2001)Google Scholar
  20. 20.
    DiBenedetto, E.: \(C^{1, \gamma }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1985)CrossRefGoogle Scholar
  21. 21.
    Fukagai, N., Narukawa, K.: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 186, 539–564 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Azzollini, A., d’Avenia, P., Pomponio, A.: Quasilinear elliptic equations in \({\mathbb{R}}^{N}\) via variational methods and Orlicz-Sobolev embeddings. Calc. Var. Partial Differ. Equ. 49, 197–213 (2014)CrossRefGoogle Scholar
  23. 23.
    Alves, C.O., da Silva, A.R.: Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space. J. Math. Phys. 57, 111502 (2016).  https://doi.org/10.1063/1.4966534 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Alves, C.O., da Silva, A.R.: Multiplicity and concentration behavior of solutions for a quasilinear problem involving N-functions via penalization method. Electron. J. Differ. Equ. 2016(158), 1–24 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Bonanno, G., Molica Bisci, G., Rădulescu, V.D.: Quasilinear elliptic non-homogeneous dirichlet problems through Orlicz-Sobolev spaces. Nonlinear Anal. 75, 4441–4456 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Bonanno, G., Molica Bisci, G., Rădulescu, V.D.: Existence and multiplicity of solutions for a quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting. J. Math. Anal. Appl. 330, 416–432 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fukagai, N., Ito, M., Narukawa, K.: Quasilinear elliptic equations with slowly growing principal part and critical Orlicz-Sobolev nonlinear term. Proc. R. Soc. Edinburgh Sect. A 139, 73–106 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Le, V.K., Motreanu, D., Motreanu, V.V.: On a non-smooth eigenvalue problem in Orlicz-Sobolev spaces. Appl. Anal. 89, 229–242 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mihailescu, M., Rădulescu, V., Repovš, D.: On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting. J. Math. Pures Appl. 93, 132–148 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rădulescu, V.D.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rădulescu, V.D., Repovš, D.: Partial Differential Equations with Variable Exponents Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC, Boca Raton (2015)CrossRefGoogle Scholar
  33. 33.
    Repovš, D.: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. 13, 645–661 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Santos, J.A.: Multiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear terms. Electron. J. Differ. Equ. 249(2013), 1–13 (2013)MathSciNetGoogle Scholar
  35. 35.
    Santos, J.A., Soares, S.H.M.: Radial solutions of quasilinear equations in Orlicz-Sobolev type spaces. J. Math. Anal. Appl. 428, 1035–1053 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Adams, A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic, Oxford (2003)zbMATHGoogle Scholar
  37. 37.
    Rao, M.N., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1985)Google Scholar
  38. 38.
    Donaldson, T.K., Trudinger, N.S.: Orlicz-Sobolev spaces and embedding theorems. J. Funct. Anal. 8, 52–75 (1971)CrossRefGoogle Scholar
  39. 39.
    Chabrowski, J.: Variational Methods for Potential Operator Equations with Applications to Nonlinear Elliptic Equations. Walter de Gruyter, Berlin-New York (1997)CrossRefGoogle Scholar
  40. 40.
    Dal Maso, G., Murat, F.: Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal. 31, 405–412 (1998)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Gossez, J.P.: Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems. In: Nonlinear Analysis, Function Spaces and Applications (Proc. Spring School, Horni Bradlo, 1978). Teubner, Leipzig, pp. 59–94 (1979)Google Scholar

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Unidade Acadêmica de MatemáticaUniversidade Federal de Campina GrandeCampina GrandeBrazil
  2. 2.Faculty of Applied MathematicsAGH University of Science and TechnologyKrakówPoland
  3. 3.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  4. 4.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  5. 5.Centro de Ciências e TecnologiaUniversidade Federal do CaririJuazeiro do NorteBrazil

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