Nodal properties of eigenfunctions of a generalized buckling problem on   balls
  In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: \begin{equation*}   \begin{cases}   \Delta^2 u+ \kappa^2 u=-\lambda \Delta u &\text{in } B_1,\newline   u=\partial_r u= 0 &\text{on } \partial B_1,   \end{cases} \end{equation*} 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 $\kappa \in (0,+\infty)$, we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction. 
Colette De Coster, Serge Nicaise, Christophe Troestler
math.AP