Laplacian eigenvalue problem

Question

I'm working through a PDE problem and we are given the eigenvalue problem $-\Delta u = \lambda u$ with $\frac{\partial u}{\partial n} = 0$ along the boundary given by the rectangle $\Omega = (0, \pi)$ x $(0, \pi)$, where $n$ is the outward normal. Using divergence theorem I can show that $\int_{C} \lambda \frac{\partial u}{\partial n} = 0$, but I'm not sure how that helps me...