Nodal properties of eigenfunctions of a generalized buckling problem on balls

TL;DR

This paper investigates the eigenfunctions of a fourth-order buckling problem on balls, revealing how the eigenvalues and eigenfunctions' properties change with the parameter , including simplicity and nodal regions.

Contribution

It provides a detailed analysis of the nodal properties and multiplicity of the first eigenvalue for a generalized buckling problem depending on a parameter .

Findings

First eigenvalue is simple and eigenfunction positive for small .

Eigenvalue multiplicity and eigenfunction sign change depend on .

Exact number of nodal regions of the first eigenfunction is characterized.

Abstract

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