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 $\Om$ and $w = 0$ on part of $\partial \Omega$. 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 $\{\rho \in L^\infty(\Omega) \colon \mathcal{A} + \rho \text{ has simple eigenvalues}\}$ forms a residual subset within $(L^\infty(\Omega), |\cdot|_\infty)$ still holds for the degenerate elliptic differential operator $\mathcal{A}$.