Let $\Omega \subset \mathbb{R}^N$ be a bounded, connected, open, and regular set. Let $u \in C^\infty(\Omega)$, such that
$$u=0, \mbox{ on }\partial \Omega.$$
Let us suppose that as a consequence of an application of the strong maximum principle for the operator $-\Delta + M \mbox{ Id}$, where $M\geq 0$, we have
$$u>0 \mbox{ in }\Omega,\;\;\; \frac{\partial u}{\partial n}<0\mbox{ on } \partial \Omega.$$
(to be more precise, u is a nontrivial supersolution for the operator $-\Delta+M\mbox{ Id}$).
Prove that there exists $C_1,C_2>0$ such that $$0<C_1 d(x,\partial \Omega)\leq u(x)\leq C_2 d(x,\partial \Omega),\; \forall x\in \Omega.$$
This result is trivial if we consider a compact set $K\subset \Omega$. My problem is how to prove this result near the boundary.