Some Key Properties of Eigenfunctions Linked to Degenerate Elliptic Differential Operators

In this study, we address the eigenvalue problem given by: \begin{equation*} \begin{cases} -\Div (w\nabla u_i)=\la_iu_i &$\text{in }$ \Om\subset $\mathbb{R}$^n,\ u_i=0 &$\text{on }$ \pt \Om, \end{cases} \end{equation*} where $w>0$ within $\text{Om}$ and $w=0$ on part of $\mathrm{\partial }\mathrm{\Omega }$. We establish Courant’s nodal domain theorem for the corresponding degenerate elliptic differential operator $\mathcal{A}$. Unlike uniformly elliptic operators, degenerate cases often result in the loss of many advantageous properties. Despite this, we show that the essential property that the set $\{\rho \in L^{\mathrm{\infty }}(\mathrm{\Omega })\text{\hspace{0.17em}⁣}:\mathcal{A}+\rho \text{ has simple eigenvalues}\}$ forms a residual subset within $(L^{\mathrm{\infty }}(\mathrm{\Omega }),\mathrm{\mid }\cdot {\mathrm{\mid }}_{\mathrm{\infty }})$ still holds for the degenerate elliptic differential operator $\mathcal{A}$.