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Positivity

, Volume 19, Issue 4, pp 843–875 | Cite as

Nodal properties of eigenfunctions of a generalized buckling problem on balls

  • Colette De Coster
  • Serge Nicaise
  • Christophe Troestler
Article

Abstract

In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates:
$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u+ \kappa ^2 u=-\lambda \Delta u &{}\text {in } B_1,\\ u=\partial _r u= 0 &{}\text {on } \partial B_1, \end{array}\right. } \end{aligned}$$
where \(B_1\) is the unit ball in \({\mathbb R}^N\). When \(\kappa > 0\) is small, we show that the first eigenvalue is simple and the first eigenfunction, which gives the shape of the film for small displacements, is positive. However, when \(\kappa \) increases, we establish that the first eigenvalue is not always simple and the first eigenfunction may change sign. More precisely, for any \(\kappa \in \mathopen ]0,+\infty \mathclose [\), we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction.

Keywords

Fourth order problem Buckling Nodal properties of eigenfunctions 

Mathematics Subject Classification

35K55 35B65 

Notes

Acknowledgments

The authors would like to thank the anonymous referee for her/his careful reading of the manuscript.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Colette De Coster
    • 1
  • Serge Nicaise
    • 1
  • Christophe Troestler
    • 2
  1. 1.Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de ValenciennesValenciennes Cedex 9France
  2. 2.Département de MathématiqueUniversité de MonsMonsBelgium

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