Spectral analysis of a generalized buckling problem on a ball
  In this paper, the spectrum of the following fourth order problem \begin{equation*}   \begin{cases}   \Delta^2 u+\nu u=-\lambda \Delta u &\text{in } D_1,\newline   u=\partial_r u= 0 &\text{on } \partial D_1,   \end{cases} \end{equation*} where $D_1$ is the unit ball in ${\mathbb R}^N$, is determined for $\nu < 0$ as well as the nodal properties of the corresponding eigenfunctions. In particular, we show that the first eigenvalue is simple and that the corresponding eigenfunction is radial and (up to a multiplicative factor) positive and decreasing with respect to the radius. This completes earlier results obtained for $\nu \ge 0$ and for $\nu <0$. 
Colette De Coster, Serge Nicaise, Christophe Troestler
math.AP