Nodal properties of eigenfunctions of a generalized buckling problem on balls

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In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u+ \kappa ^2 u=-\lambda \Delta u &{}\text {in } u=\partial _r u= 0 &{}\text {on } \partial B_1, \end{array}\right. } \end{aligned}$$ 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.