Advertisement

Monatshefte für Mathematik

, Volume 190, Issue 4, pp 615–639 | Cite as

Infinitely many periodic solutions for a class of fractional Kirchhoff problems

  • Vincenzo AmbrosioEmail author
Article

Abstract

We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schrödinger operator with periodic boundary conditions and involving different types of nonlinearities.

Keywords

Periodic solutions Fractional Kirchhoff equation Variational methods Critical exponent 

Mathematics Subject Classification

34K13 35R11 35A15 35B33 

Notes

Acknowledgements

The author would like to express his sincere gratitude to the referee for all insightful comments and valuable suggestions, which enabled to improve this version of the manuscript.

References

  1. 1.
    Alves, C.O., Corrêa, F.J.S.A., Figueiredo, G.M.: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2(3), 409–417 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49(1), 85–93 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alves, C.O., Figueiredo, G.M.: Nonlinear perturbations of a periodic Kirchhoff equation in \(\mathbb{R}^{N}\). Nonlinear Anal. 75(5), 2750–2759 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ambrosio, V.: Periodic solutions for a pseudo-relativistic Schrödinger equation. Nonlinear Anal. TMA 120, 262–284 (2015)zbMATHGoogle Scholar
  6. 6.
    Ambrosio, V.: Periodic solutions for the non-local operator pseudo-relativistic \((-\Delta +m^{2})^{s}-m^{2s}\) with \(m\ge 0\). Topol. Methods Nonlinear Anal. 49(1), 75–104 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ambrosio, V.: Periodic solutions for a superlinear fractional problem without the Ambrosetti–Rabinowitz condition. Discrete Contin. Dyn. Syst. 37(5), 2265–2284 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ambrosio, V.: Periodic solutions for critical fractional equations. Calc. Var. Partial Differ. Equ. 57(2), 57:45 (2018)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ambrosio, V., Isernia, T.: A multiplicity result for a fractional Kirchhoff equation in \(\mathbb{R} ^{N}\) with a general nonlinearity. Commun. Contemp. Math. 20(5), 1750054 (2018). 17 ppMathSciNetzbMATHGoogle Scholar
  10. 10.
    Ambrosio, V., Isernia, T.: Concentration phenomena for a fractional Schrödinger–Kirchhoff type problem. Math. Methods Appl. Sci. 41(2), 615–645 (2018)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ambrosio, V., Mawhin, J., Molica Bisci, G.: (Super)Critical nonlocal equations with periodic boundary conditions. Selecta Math. (N.S.) 24(4), 3723–3751 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Barrios, B., Colorado, E., de Pablo, A., Sánchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252(11), 6133–6162 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Benci, V.: On critical points theory for indefinite functionals in the presence of symmetric. Trans. Am. Math. Soc. 274, 533–572 (1982)zbMATHGoogle Scholar
  15. 15.
    Bernstein, S.: Sur une classe d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4, 17–26 (1940)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondracěk, Z.: Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Mathematics, vol. 1980. Springer, Berlin (2009)Google Scholar
  17. 17.
    Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30(7), 4619–4627 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Clark, D.C.: A variant of the Lusternik–Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972/1973)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of \(\mathbb{R}^{n}\), Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa. viii+152 pp (2017)Google Scholar
  24. 24.
    Figueiredo, G.M., Molica Bisci, G., Servadei, R.: On a fractional Kirchhoff-type equation via Krasnoselskii’s genus. Asymptot. Anal. 94(3–4), 347–361 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Figueiredo, G.M., Santos, J.R.: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integral Equ. 25(9–10), 853–868 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Garcia Azorero, J., Peral Alonso, I.: Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Am. Math. Soc. 323, 877–895 (1991)MathSciNetzbMATHGoogle Scholar
  28. 28.
    He, Y., Li, G., Peng, S.: Concentrating bound states for Kirchhoff type problems in \(\mathbb{R}^{3}\) involving critical Sobolev exponents. Adv. Nonlinear Stud. 14(2), 483–510 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Hebey, E.: Multiplicity of solutions for critical Kirchhoff type equations. Commun. Partial Differ. Equ. 41(6), 913–924 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Hebey, E., Thizy, P.-D.: Stationary Kirchhoff systems in closed \(3\)-dimensional manifolds. Calc. Var. Partial Differ. Equ. 54(2), 2085–2114 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)zbMATHGoogle Scholar
  32. 32.
    Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lieb, E.H., Loss, M.: Analysis, Vol. 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  34. 34.
    Lions, J.L.: On some questions in boundary value problems of mathematical physics, contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), pp. 284–346, North-Holland Math. Stud., 30, North-Holland, Amsterdam (1978)Google Scholar
  35. 35.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1, 45–121 (1985)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Liu, J., Liao, J.-F., Tang, C.-L.: Positive solutions for Kirchhoff-type equations with critical exponent in \(\mathbb{R}^N\). J. Math. Anal. Appl. 429(2), 1153–1172 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational methods for nonlocal fractional problems, vol. 162. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  38. 38.
    Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18(2), 489–502 (2013)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221(1), 246–255 (2006)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Pohožaev, S.I.: A certain class of quasilinear hyperbolic equations. Mat. Sb. 96, 152–166 (1975)MathSciNetGoogle Scholar
  42. 42.
    Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^{N}\). Calculus Var. PDE 54, 2785–806 (2015)zbMATHGoogle Scholar
  43. 43.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Confer. Ser. Math. 65, 58 (1986)MathSciNetGoogle Scholar
  44. 44.
    Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35, 2092–2122 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

Personalised recommendations