# Nodal solutions for a nonlinear fourth-order eigenvalue problem

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DOI: 10.1007/s10114-007-1009-6

- Cite this article as:
- Ma, R.Y. & Thompson, B. Acta. Math. Sin.-English Ser. (2008) 24: 27. doi:10.1007/s10114-007-1009-6

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## Abstract

We are concerned with determining the values of where

*λ*, for which there exist nodal solutions of the fourth-order boundary value problem$$
\begin{gathered}
y'''' = \lambda a(x)f(y),0 < x < 1, \hfill \\
y(0) = y(1) = y''(0) = y'' = (1) = 0, \hfill \\
\end{gathered}
$$

*λ*is a positive parameter,*a*∈*C*([0, 1], (0, ∞)),*f*∈*C*(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 resultseigenvaluesbifurcation methodsnodal zeros### MR(2000) Subject Classification

34B15## Copyright information

© Springer-Verlag 2008