Nodal properties of eigenfunctions of a generalized buckling problem on balls
C. De Coster, Serge Nicaise, Christophe Troestler · arXiv (Cornell University) · 2014
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} Δ^2 u+ κ^2 u=-λΔu &\text{in } B_1, ewline 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 $κ> 0$ 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 $κ$ increases, we establish that the first eigenvalue is not always simple and the first eigenfunction may change sign. More precisely, for any $κ\in (0,+\infty)$, we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction.