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:

Δ² u+ κ² u=-λ Δ u in B₁,
u=∂ᵣu= 0 on ∂B_1,

where B₁ is the unit ball in Rᴺ. 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 ∈]0, +∞[, we give the exact multiplicity of the
first eigenvalue and the number of nodal regions of the first
eigenfunction.