Acta Mathematica Sinica, English Series

, Volume 24, Issue 1, pp 27–34

Nodal solutions for a nonlinear fourth-order eigenvalue problem



We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem
$$ \begin{gathered} y'''' = \lambda a(x)f(y),0 < x < 1, \hfill \\ y(0) = y(1) = y''(0) = y'' = (1) = 0, \hfill \\ \end{gathered} $$
where λ is a positive parameter, aC([0, 1], (0, ∞)), fC (ℝ, ℝ) satisfies f(u)u > 0 for all u ≠ 0. We give conditions on the ratio f(s)/s, at infinity and zero, that guarantee the existence of nodal solutions. The proof of our main results is based upon bifurcation techniques.


multiplicity results eigenvalues bifurcation methods nodal zeros 

MR(2000) Subject Classification



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  1. [1]
    Henderson, J., Wang, H. Y.: Positive solutions for nonlinear eigenvalue problems. J. Math. Anal. Appl., 208, 252–259 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Naito, Y., Tanaka, S.: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations. Nonlinear Analysis TMA 56(4), 919–935 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal., 7, 487–513 (1971)MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Ruf, B.: Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna. Nonlinear Analysis TMA, 9(12), 1325–1330 (1985)MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Bai, Z. B., Wang, H. Y.: On positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl., 270(2), 357–368 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Del Pino, M. A., Mansevich, R. F.: Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition. Proc. Amer. Math. Soc., 112(1), 81–86 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Ding, T. R., Zanolin, F.: Periodic solutions of Duffing’s equations with superquadratic potential. J. Differential Equations, 97(2), 328–378 (1992)MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Ma, R. Y.: Bifurcation from infinity and multiple solutions for periodic boundary value problems. Nonlinear Anal. TMA, 42(1), 27–39 (2000)CrossRefGoogle Scholar
  9. [9]
    Ma, R. Y., Wang, H. Y.: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal., 59(1–4), 225–231 (1995)MATHMathSciNetGoogle Scholar
  10. [10]
    Ma, R. Y., Ma, Q. Z.: Positive solutions for semipositone m-point boundary-value problems. Acta Mathematica Sinica, English Series, 20(2), 273–282 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Li, W. T., Sun, H. R.: Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales. Acta Mathematica Sinica, English Series, 22(6), 1797–1804 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Bai, Z. B., Ge, W. G.: Existence of positive solutions to fourth order quasilinear boundary value problems.Acta Mathematica Sinica, English Series, 22(6), 1825–1830 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Stanek, S.: Existence of positive solutions to semipositone singular Dirichlet boundary value problems. Acta Mathematica Sinica, English Series, 22(6), 1891–1914 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Yao, Q. L.: Existence, Multiplicity and Infinite Solvability of Positive Solutions for One-Dimensional p-Laplacian. Acta Mathematica Sinica, English Series, 21(4), 691–698 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Rynne, B. P.: Infinitely many solutions of superlinear fourth order boundary value problems. Topol. Methods Nonlinear Anal., 19(2), 303–312 (2002)MATHMathSciNetGoogle Scholar
  16. [16]
    Elias, U.: Eigenvalue problems for the equation Ly + λp(x)y = 0. J. Differential Equations, 29, 28–57 (1978)CrossRefMathSciNetGoogle Scholar
  17. [17]
    Lazer, A. C., McKenna, P. J.: Global bifurcation and a theorem of Tarantello. J. Math. Anal. Appl., 181(3), 648–655 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Rynne, B. P.: Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems. J. Differential Equations, 188, 461–472 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouP. R. China
  2. 2.Department of Mathematicsthe University of QueenslandBrisbaneAustralia

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