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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, 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.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P. R. China

    Ru Yun Ma

  2. Department of Mathematics, the University of Queensland, Brisbane, Qld, 4072, Australia

    Bevan Thompson

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  1. Ru Yun Ma
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  2. Bevan Thompson
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Correspondence to Ru Yun Ma.

Additional information

Supported by the NSFC (No. 10671158), the NSF of Gansu Province (No. 3ZS051-A25-016), NWNUKJCXGC-03-17, the Spring-Sun Program (No. Z2004-1-62033), SRFDP (No. 20060736001), and the SRF for ROCS, SEM (2006[311])

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Ma, R.Y., Thompson, B. Nodal solutions for a nonlinear fourth-order eigenvalue problem. Acta. Math. Sin.-English Ser. 24, 27–34 (2008). https://doi.org/10.1007/s10114-007-1009-6

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  • DOI: https://doi.org/10.1007/s10114-007-1009-6

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