Two solutions for a fourth order nonlocal problem with indefinite potentials

  • Giovany M. FigueiredoEmail author
  • Marcelo F. Furtado
  • João Pablo P. da Silva


We study the nonlocal equation
$$\begin{aligned} \Delta ^{2}u-m\left( \displaystyle \int _{\Omega }|\nabla u|^{2} dx \right) \Delta u = \lambda a(x) |u|^{q-2}u+ b(x)|u|^{p-2}u, \, \text{ in } \Omega , \end{aligned}$$
subject to the boundary condition \(u=\Delta u=0\) on \(\partial \Omega \). For m continuous and positive we obtain a nonnegative solution if \(1<q<2<p \le 2N/(N-4)\) and \(\lambda >0\) small. If the affine case \(m(t)=\alpha +\beta t\), we obtain a second solution if \(4<p<2N/(N-4)\) and \(N \in \{5,6,7\}\). In the proofs we apply variational methods.

Mathematics Subject Classification

Primary 35J60 Secondary 35J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alama, S., Tarantello, G.: On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. PDE 1, 439–475 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ashfaque, A.B., Mahomed, F.M., Zaman, F.D.: Invariant boundary value problems for a fourth-order dynamic Euler–Bernoulli beam equation. J. Math. Phys. 53, 043703 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhakta, M., Mukherjee, D.: Multiplicity results and sign changing solutions of non-local equations with concave–convex nonlinearities. Differ. Int. Equ. 30, 387–422 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berger, M.: A new approach to the large deflection of plate. J. Appl. Mech. 22, 465–472 (1955)MathSciNetGoogle Scholar
  6. 6.
    Bernis, F., García Azorero, J., Peral, I.: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1, 219–240 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boccardo, L., Escobedo, M., Peral, I.: A Dirichlet problem involving critical exponents. Nonlinear Anal. 24, 1639–1648 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chueshov, I., Lasiecka, I.: Long-time behavior of second order evolution equations with nonlinear damping. Memoirs of the American Mathematical Society, vol. 165. American Mathematical Society, Providence, RI (2008)Google Scholar
  10. 10.
    de Figueiredo, D.G., Gossez, J.P., Ubilla, P.: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 199, 452–467 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Drábek, P., Huang, Y.X.: Multiplicity of positive solutions for some quasilinear elliptic equation in \(\mathbb{R}^N\) with critical Sobolev exponent. J. Differ. Equ. 140, 106–132 (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Figueiredo, G.M., Nascimento, R.G.: Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator. Electron. J. Differ. Equ., paper no. 217 (2016)Google Scholar
  14. 14.
    Kang, J.R.: Energy decay of solutions for an extensible beam equation with a weak nonlinear dissipation. Math. Methods Appl. Sci. 35, 1587–1593 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liao, J.-F., Pu, Y., Ke, X.-F., Tang, C.-L.: Multiple positive solutions for Kirchhoff type problems involving concave–convex nonlinearities. Commun. Pure Appl. Anal. 16, 2157–2175 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, T.-F., Narciso, V.: Global attractor for a model of extensible beam with nonlinear damping and source terms. Nonlinear Anal. 73, 3402–3412 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mao, A., Wang, W.: Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in \(\mathbb{R}^{3}\). J. Math. Anal. Appl. 459, 556–563 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Melo, J.L.F., dos Santos, E.M.: Positive solutions to a fourth-order elliptic problem by the Lusternik–Schnirelmann category. J. Math. Anal. Appl. 420, 532–550 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Naimen, D.: The critical problem of Kirchhoff type elliptic equations in dimension four. J. Differ. Equ. 257, 1168–1193 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pádua, J.C.N., Silva, E.A.B., Soares, S.H.M.: Positive solutions of critical semilinear problems involving a sublinear term on the origin. Indiana Univ. Math. J. 55, 1091–1111 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Song, H., Chen, C.: Infinitely many solutions for Schrödinger–Kirchhoff-type fourth-order elliptic equations. Proc. Edinb. Math. Soc. 60, 1003–1020 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Song, Y., Shi, S.: Multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent. J. Dyn. Control Syst. 23, 375–386 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tokens, E.: A semilinear elliptic equation with convex and concave nonlinearities. Top. Methods Nonlinear Anal. 13, 251–271 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Van der Vorst, R.C.A.M.: Fourth-order elliptic equations with critical growth. C. R. Math. Acad. Sci. Paris 320, 295–299 (1995)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Woinowsky-Krieger, S.: The effect of axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wu, T.F.: Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight. J. Differ. Equ. 249, 1549–1578 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, F., Avci, M., An, A.: Existence of solutions for fourth order elliptic equations of Kirchhoff type. J. Math. Anal. Appl. 401(7), 140–146 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yang, Z.-J.: On an extensible beam equation with nonlinear damping and source terms. J. Differ. Equ. 254, 3903–3927 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Giovany M. Figueiredo
    • 1
    Email author
  • Marcelo F. Furtado
    • 1
  • João Pablo P. da Silva
    • 2
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal do ParáSoureBrazil

Personalised recommendations