Positivity

pp 1–22 | Cite as

Spectral analysis of a generalized buckling problem on a ball

  • Colette De Coster
  • Serge Nicaise
  • Christophe Troestler
Article

Abstract

In this paper, the spectrum of the following fourth order problem
$$\begin{aligned} {\left\{ \begin{array}{ll} \varDelta ^2 u+\nu u=-\lambda \varDelta u &{}\quad \text {in } D_1,\\ u=\partial _r u= 0 &{}\quad \text {on } \partial D_1, \end{array}\right. } \end{aligned}$$
where \(D_1\) is the unit ball in \({\mathbb {R}}^N\), is determined for \(\nu < 0\) as well as the nodal properties of the corresponding eigenfunctions. In particular, we show that the first eigenvalue is simple and that the corresponding eigenfunction is radial and (up to a multiplicative factor) positive and decreasing with respect to the radius. This completes earlier results obtained for \(\nu \geqslant 0\) (see Coster et al. in Positivity 19:843–875, 2015) and for \(\nu <0\) (see Laurençot and Walker in J Anal Math 127:69–89, 2014).

Keywords

Buckling problem Spectral analysis Properties of eigenfunctions 

Mathematics Subject Classification

35J40 (35B65 35J35 35P15) 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

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

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