So, this seens to be simple but I could not handle to prove it:
Suppose $\Omega$ is a bounded and open domain of $\mathbb R^N$ with smooth boundary (as smooth as you want), $N\geq 2$ and that $u\in C^1(\overline\Omega,\mathbb R)$ is a function such that $\partial u/\partial\nu<0$ all over $\partial\Omega$, where $\nu$ denotes the outward normal unit vector on $\partial \Omega$.
Is it true that $u>0$ near the boundary? That means: can a $\delta>0$ be chosen such that $u(x)>0$ for all $x\in\{y\in\Omega;\ dist(y,\partial\Omega)<\delta\}$?