球上广义屈曲问题的谱分析
Spectral analysis of a generalized buckling problem on a ball
摘要：
本文确定了以下四阶问题\begin{equation*} \begin{cases} Δ^2 u+νu=-λΔu &\text{in } D_1,\newline u=\partial_r u= 0 &\text{on } \partial D_1, \end{cases} \end{equation*}的谱，其中$D_1$是${\mathbb R}^N$中的单位球，并给出了$ν< 0$的本征函数的节点性质.特别是，我们表明，第一特征值是简单的，相应的本征函数是径向和（乘法因子）积极和减少的半径。这完成了先前对$ν\ge 0$和$ν<0$获得的结果。
英文摘要：
In this paper, the spectrum of the following fourth order problem \begin{equation*} \begin{cases} Δ^2 u+νu=-λΔu &\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 $ν< 0$ 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 $ν\ge 0$ and for $ν<0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
