Spectral analysis of a generalized buckling problem on a ball

TL;DR

This paper analyzes the spectral properties of a fourth-order buckling problem on a unit ball, determining eigenvalues, eigenfunctions, and their properties for negative and non-negative parameters, extending previous results.

Contribution

It provides a complete spectral analysis of a generalized buckling problem on a ball, including eigenvalues, eigenfunctions, and their nodal properties for all relevant parameter ranges.

Findings

First eigenvalue is simple and eigenfunction is radial and decreasing.

Spectral properties are fully characterized for negative and non-negative parameters.

Extends previous results to include the case of negative parameter a5.

Abstract

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 $\nu <0$.