Multiplicity of high energy solutions for superlinear Kirchhoff equations

Original Research


In this paper, we study the existence of infinitely many high energy solutions for the nonlinear Kirchhoff equations
$$\left\{\everymath{\displaystyle}\begin{array}{l@{\quad}l}- \biggl(a+b\int_{R^3} |\nabla u|^2 dx\biggr)\Delta u + V(x)u=f(x,u),&x\in \mathbb {R}^3,\\[9pt]u\in H^1 (\mathbb {R}^3),\end{array}\right.$$
where a,b>0 are constants, V:ℝ3→ℝ is continuous and has a positive infimum. f is a subcritical nonlinearity which needs not to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions.


Nonlinear Kirchhoff equations High energy solutions Variational methods 

Mathematics Subject Classification (2000)

35J60 35J25 



The authors are grateful for the anonymous referees for very helpful suggestions and comments. This work was supported by NSFC Grants 10971238 and the Fundamental Research Funds for the Central Universities 0910KYZY51.


  1. 1.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883) Google Scholar
  2. 2.
    Bernstein, S.: Sur une class d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS, Ser. Math., Izv. Akad. Nauk SSSR 4, 17–26 (1940) Google Scholar
  3. 3.
    Pohoẑaev, S.I.: A certain class of quasilinear hyperbolic equations. Mat. Sb. 96, 152–168 (1975) MathSciNetGoogle Scholar
  4. 4.
    Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1997. North-Holland Math. Stud., vol. 30, pp. 284–346 (1978) CrossRefGoogle Scholar
  5. 5.
    Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001) MathSciNetMATHGoogle Scholar
  7. 7.
    Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30, 4619–4627 (1997) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    D’Ancona, A.P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    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, 85–93 (2005) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ma, T.F., Muñoz Rivera, J.E.: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 16, 243–248 (2003) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Zhang, Z., Perera, K.: Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mao, A., Zhang, Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    He, X., Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Cheng, B., Wu, X.: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal. 71, 4883–4892 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in ℝN. Nonlinear Anal., Real World Appl. 12, 1278–1287 (2011) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ricceri, B.: On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim. 46, 543–549 (2010) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Heidarkhani, S., Afrouzi, G.A., O’Regan, D.: Existence of three solutions for a Kirchhoff-type boundary value problem. Electron. J. Differ. Equ. 2011(91), 1–11 (2011) MathSciNetGoogle Scholar
  19. 19.
    Heidarkhani, S., Tian, Y.: Three solutions for a class of gradient Kirchhoff-type systems depending on two parameters. Dyn. Syst. Appl. 20, 551–562 (2011) MathSciNetGoogle Scholar
  20. 20.
    Bartsch, T., Pankov, A., Wang, Z.Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Comtemp. Math. 3, 549–569 (2001) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Bartsch, T., Wang, Z.Q.: Existence and multiple results for some superlinear elliptic problems on R N. Commun. Partial Differ. Equ. 20, 1725–1741 (1995) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) MATHCrossRefGoogle Scholar
  24. 24.
    Zou, W.: Variant fountain theorem and their applications. Manuscr. Math. 104, 343–358 (2001) MATHCrossRefGoogle Scholar
  25. 25.
    Zou, W., Schechter, M.: Critical Point Theory and Its Applications. Springer, New York (2006) MATHGoogle Scholar
  26. 26.
    He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ3. J. Differ. Equ. (2011). doi:10.1016/j.jde.2011.08.035 Google Scholar
  27. 27.
    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. American Mathematical Society, Providence (1986) Google Scholar
  28. 28.
    Jeajean, L.: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ℝN. Proc. R. Soc. Edinb. A 129, 787–809 (1999) CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2012

Authors and Affiliations

  1. 1.College of ScienceMinzu University of ChinaBeijingP.R. China

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