Let $\displaystyle D$ be the unit disk in $\displaystyle \mathbb{R}^2$ and let $\displaystyle \partial D$ denote the boundary. Show that the only solution of the boundary value problem:

$\displaystyle \Delta u - u = 0$ in $\displaystyle D$

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

is $\displaystyle u=0$. The hint given is to apply the 2-D divergence theorem on the vector field $\displaystyle u\nabla u$, but I don't see how that even enters in.