Mathematics

CLC number: O231.4

The Eigenvalue Properties of a Kind of Singular Differential Equations in 3-Dimensional Space

School of Mathematics and Statistics, Central South University, Changsha 410000, Hunan, China

^† Corresponding author. E-mail: ghadirshokor@gmail.com

Received: 10 June 2023

Abstract

In this paper, we consider the eigenvalue problem of the singular differential equation $-\mathrm{\Delta }{u}_i-\frac{h}{{\left|x\right|}^{\mathrm{2}}}{u}_i+V(x)u_i={\lambda }_i(V,h)u_i$ in a bounded open ball with Dirichlet boundary condition in 3-dimensional space, where, $V\in {V}=\{a\in L^{\mathrm{\infty }}(\mathrm{\Omega })|\mathrm{0}\le a\le M$$\mathrm{a}.\mathrm{e}.$, $M$ is a given constant$\}$. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.