maximum principle on compact manifolds with boundary

Question

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ gives a nonlinearity, $f$ being an odd function. I was wondering if $u \geq 0$ on $M$ would mean either $u > 0$ on $M$ or $u = 0$ on $\overline{M}$. I could use a reference wherein I can look up the proof.