Let $\displaystyle U\subset \mathbb{R}^n$ be a smooth bounded region (open, connected with smooth boundary). Assume $\displaystyle u\in C^2(U) \cap C(\overline{U})$ solves

$\displaystyle -\Delta u = \lambda u \ \ \ \ in \ \ U$

$\displaystyle \ \ \ \ u=0 \ \ \ \ \ on \ \ \partial U$

with $\displaystyle \lambda \in \mathbb{R}$. Prove that if $\displaystyle u$ is not identically $\displaystyle 0$ then $\displaystyle \lambda >0$.

Note: Justify all your arguments.