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 $\kappa \in (0,+\mathrm{\infty })$, we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction.