  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.
