Acta Mathematica Sinica, English Series

, Volume 24, Issue 1, pp 27–34

Nodal solutions for a nonlinear fourth-order eigenvalue problem

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Abstract

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.

Keywords

multiplicity results eigenvalues bifurcation methods nodal zeros 

MR(2000) Subject Classification

34B15 

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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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