On a nonlocal problem involving a nonstandard nonhomogeneous differential operator

  • Mustafa AvciEmail author
  • Berat Süer


In this article, we are concerned with some class of nonlocal problems involving a nonstandard nonlocal and nonhomogeneous differential operator settled in Musielak–Orlicz–Sobolev spaces. We apply variational approach and look for nontrivial solutions, that is, local minimizers of the corresponding energy functional.


Nonlocal problems Ginzburg–Landau energy Variable exponent Variational approach Mountain-Pass theorem Musielak–Orlicz–Sobolev spaces 

Mathematics Subject Classification

35J60 35J92 58E05 



  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic, New York (1975)zbMATHGoogle Scholar
  2. 2.
    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 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avci, M., Cekic, B., Mashiyev, R.A.: Existence and multiplicity of the solutions of the \(p(x)\)-Kirchhoff type equation via genus theory. Math. Methods Appl. Sci. 34(14), 1751–1759 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Avci, M., Pankov, A.: Multivalued elliptic operators with nonstandard growth. Adv. Nonlinear Anal. 7(1), 35–48 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ayazoglu (Mashiyev), R., Avci, M., Chung, N.T.: Existence of solutions for nonlocal problems in Sobolev–Orlicz spaces via monotone method. Electron. J. Math. Anal. Appl. 4(1), 63–73 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bethuel, F., Brezis, H., Hlein, F.: Ginzburg–Landau Vortices. Birkhuser, Basel (1994)CrossRefGoogle Scholar
  8. 8.
    Bethuel, F., Brezis, H., Hlein, F.: Asymptotics for the minimization of a Ginzburg–Landau functional. Calc Var PDE 1, 123–148 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bonanno, G., Molica Bisci, G., Rădulescu, V.: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces. Nonlinear Anal. TMA 75, 4441–4456 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Boureanu, M.M., Udrea, D.N.: Existence and multiplicity results for elliptic problems with \(p(.)\)-Growth conditions. Nonlinear Anal. Real World Appl. 14(4), 1829–1844 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chipot, M.: Elliptic Equations: An Introductory Course. Birkhuser, Basel (2009)CrossRefGoogle Scholar
  12. 12.
    Colasuonno, F., Pucci, P.: Multiplicity of solutions for \(p(x)\)-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Corréa, F.J.S.A., Figueiredo, G.M.: On a \(p\)-Kirchhoff equation via Krasnoselskiis genus. Appl. Math. Lett. 22, 819–822 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Comte, M., Mironescu, P.: Minimizing properties of arbitrary solutions to the Ginzburg–Landau equation. Proc. R. Soc. Edinburgh Sect. A Math. 129(6), 1157–1169 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer, Basel (2013)CrossRefGoogle Scholar
  16. 16.
    Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Fan, X.L.: Differential equations of divergence form in Musielak–Sobolev spaces and a sub-supersolution method. J. Math. Anal. Appl. 386, 593–604 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fan, X.L.: On nonlocal \(p(x)\)-Laplacian Dirichlet problems. Nonlinear Anal. 72, 3314–3323 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fang, F., Tan, Z.: Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz–Sobolev setting. J. Math. Anal. Appl. 389, 420–428 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fukagai, N., Ito, M., Narukawa, K.: Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on \(\mathbb{R}^{\mathbb{N}}\). Funkcial. Ekvac. 49, 235–267 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Galewski, M.: A new variational method for the \(p(x)\)-Laplacian equation. Bull. Aust. Math. Soc. 72(1), 53–65 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ginzburg, V., Landau, L.: On the theory of superconductivity. Zh.exper.teor.Fiz. 20, 1064–1082 (1950)Google Scholar
  25. 25.
    Harjulehto, P., Hästö, P., Klén, R.: Generalized Orlicz spaces and related PDE. Nonlinear Anal. Theory Methods Appl. 143, 155173 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hudzik, H.: On generalized Orlicz–Sobolev space. Funct. Approx. Comment. Math. 4, 37–51 (1976)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Heidarkhani, S., Caristi, G., Ferrara, M.: Perturbed Kirchhoff-type Neumann problems in Orlicz–Sobolev spaces. Comput. Math. Appl. 71, 2008–2019 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jimbo, S., Morito, Y.: Notes on the limit equation of vortex motion for the Ginzburg–Landau equation with Neumann condition. Jpn. J. Ind. Appl. Math. 18, 483–501 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)zbMATHGoogle Scholar
  30. 30.
    Krasnosels’kii, M., Rutic’kii, J.: Convex Functions and Orlicz Spaces. Noordhoff, Groningen (1961)Google Scholar
  31. 31.
    Kufner, A., John, O., Fučik, S.: Function Spaces. Noordhoff, Leyden (1977)zbMATHGoogle Scholar
  32. 32.
    Lefter, C., Rădulescu, V.D.: On the Ginzburg-Landau energy with weight. Annales de l’Institut Henri Poincar C Analyse Non Linaire 13(2), 171–184 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lu, K., Pan, X.B.: Ginzburg–Landau equation with DeGennes boundary condition. J. Differ. Equ. 129, 136–165 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mihăilescu, M., Rădulescu, V.: Neumann problems associated to non-homogeneous differential operators in Orlicz–Sobolev spaces. Ann. Inst. Fourier 58, 2087–2111 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mironescu, P.: On the stability of radial solutions of the Ginzburg–Landau equation. J. Func. Anal. 130, 334–344 (1995)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Morito, Y.: Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain. Jpn. J. Ind. Appl. 21, 129–147 (2001)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Musielak, J.: Modular Spaces and Orlicz Spaces, Lecture Notes in Math, vol. 1034. Springer, Berlin (1983)CrossRefGoogle Scholar
  38. 38.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematics Society, Providence (1986)CrossRefGoogle Scholar
  39. 39.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc., New York (1991)zbMATHGoogle Scholar
  40. 40.
    Rădulescu, V.D., Repovs̆, D.D.: Partial Differential Equations with Variable Equations: Variational Methods and Qualitative Analysis. CRC, New York (2015)CrossRefGoogle Scholar
  41. 41.
    Struwe, M.: Une estimation asymptotique pour le modle de Ginzburg–Landau. C. R. Acad. Sci. Paris 317, 677–680 (1993)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Yucedag, Z.: Existence of solutions for \(p(x)\)-Laplacian equations without Ambrosetti–Rabinowitz type condition. Bull. Malays. Math. Sci. Soc. 38(3), 1023–1033 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Orthogonal Publisher and Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Economics and Administrative SciencesBatman UniversityBatmanTurkey
  2. 2.Graduate School of Natural and Applied SciencesBatman UniversityBatmanTurkey

Personalised recommendations