Suppose that $\Omega \subseteq \mathbb{R}^n$ is a bounded domain and $u : \Omega \to \mathbb{C}$ solves $-\Delta u = \lambda u$ with Dirichet or Neumann boundary conditions.
Can we say anything about the integrability of $u^{-\alpha}$ for some $\alpha > 0$?
I have seen that in the more general case of a Riemannian manifold we can have points with vanishing order of the order $\sqrt{\lambda}$. Can we hope to do better specialising to the case of a flat manifold with boundary as in the case above?
