Nontrivial solutions for a fourth-order elliptic equation of Kirchhoff type via Galerkin method

  • Fanglei Wang
  • Yuanfang Ru
  • Tianqing An


In this paper we study the existence of nontrivial solutions of the nonlocal elliptic problem
$$\begin{aligned} \left\{ \begin{array}{lcr} \Delta ^2 u-(a+b\int _\Omega |\nabla u|^2\mathrm{d}x)\Delta u+u=(\int _\Omega g(x,u)\mathrm{d}x)^\gamma f(x,u),\;in\;\Omega ,\\ u=\Delta u=0,\;\;on\;\;\partial \Omega \end{array}\right. \end{aligned}$$
via Galerkin method, where \(\Omega \subset \mathbb {R}^N (N\ge 5)\) is a smooth bounded domain.


Nonlocal elliptic problems biharmonic operator Galerkin method 

Mathematics Subject Classification

Primary 35J58 35J65 



The authors would like to thank the referees for the helpful suggestions.


  1. 1.
    Alves, C.O., Corrêa, F.J.S.A.: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 8, 43–56 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alves, C.O., de Figueiredo, D.G.: Nonvariational elliptic systems via Galerkin methods. In: Spaces, Function (ed.) Differential operators and nonlinear analysis-the hans triebel anniversary volume, pp. 47–57. Switzerland, Birkhäser (2003)Google Scholar
  3. 3.
    Cabada, A., Figueiredo, G.M.: A generalization of an extensible beam equation with critical growth in RN. Nonlinear Anal. Real World Appl. 20, 134–142 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Correa, F.J.S.A., Menezes, S.D.B.: Existence of solutions to nonlocal and singular elliptic problems via Galerkin method. Electron. J. Differ. Equ. 19, 1–10 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Correa, F.J.S.A., de Morais Filho, Daniel C.: On a class of nonlocal elliptic problems via Galerkin method. J. Math. Anal. Appl. 310, 177–187 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hao, X., Liu, L., Wu, Y., Xu, N.: Multiple positive solutions for singular nonlocal boundary value problems in Banach spaces. Comput. Math. Appl. 61, 1880–1890 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Heidarkhani, S., Massimiliano, F., Somaye, K.: Nontrivial solutions for one-dimensional fourth-order Kirchhoff-type equations. Mediterr. J. Math. 13, 217–236 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hao, X., Xu, N., Liu, L.: Existence and uniqueness of positive solutions for fourth-order m-point nonlocal boundary value problems with two parameters. Rocky Mountain J. Math. 43, 1161–1180 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(R^3\). J. Differ. Equ. 252, 1813–1834 (2012)CrossRefzbMATHGoogle Scholar
  10. 10.
    Khoutir, S., Chen, H.: Ground state solutions and least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in \(R^N\). Arab J. Math. Sci. 23, 94–108 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lions, J.L.: Quelques Méthodes de résolution de problémes aux limites non linéaires, Dunod. Gauthier-Villars, Paris (1969)zbMATHGoogle Scholar
  12. 12.
    Ma, T.F.: Existence results and numerical solutions for a beam equation with nonlinear boundary condition. Appl. Numer. Math. 47, 189–196 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mao, A., Chang, H.: Kirchhoff type problems in \(R^N\) with radial potentials and locally Lipschitz functional. Appl. Math. Lett. 62, 49–54 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mao, A., Luan, S.: Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems. J. Math. Anal. Appl 383, 239–243 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mao, A., Jing, R., Luan, S., Chu, J., Kong, Y.: Some nonlocal elliptic problem involing positive parameter. Topol. Methods Nonlinear Anal. 42, 207–220 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mao, A., Zhu, X.: Existence and multiplicity results for Kirchhoff problems. Mediterr. J. Math. 14, 1–14 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mao, A., Zhang, Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 270, 1275–1287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stanczy, R.: Nolocal elliptic equation. Nonliear Anal. 47, 3579–3584 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tavani, M.R.H., Afrouzi, G.A., Heidarkhani, S.: Multiplicity results for perturbed fourth-order Kirchhoff-type problems. Opusc. Math. 37, 755–772 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, F., Avci, M., An, Y.: Existence of solutions for a fourth order elliptic equation of Kirchhoff type. J. Math. Anal. Appl. 409, 140–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, F., An, T., An, Y.: Existence of solutions for a fourth order elliptic equation of Kirchhoff type on RN. Electron. J. Qual. Theory Differ. Equ. 39, 1–11 (2014)CrossRefGoogle Scholar
  23. 23.
    Wionowsky-Kreiger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)MathSciNetGoogle Scholar
  24. 24.
    Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 3599–3605 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xu, L., Chen, H.: Multiple solutions for the nonhomogeneous fourth order elliptic equations of Kirchhoff-type. Taiwanese J. Math. 19, 1215–1226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, K.: Non trivial solutions off fourth-order singular boundary value problems with sign-changing nonlinear terms. Topol. Methods Nonlinear Anal. 40, 53–70 (2012)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Depatment of Mathematics, College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of ScienceChina pharmaceutical UniversityNanjingPeople’s Republic of China

Personalised recommendations