Comparison principle

This principle is from the harmonic function theory. Let $$\Omega \subset \mathbb{R}^d$$ be a bounded, open subset of $$\mathbb{R}^d$$ with compact closure $$\overline{\Omega}$$ for some $$d \in \mathbb{N}$$ and $$u \in C^2(\Omega) \cap C(\overline{\Omega})$$ (function of class $$C^2$$ and $$C$$ on $$\Omega$$ and $$\overline{\Omega}$$, respectively). Given $$c \in C(\Omega)$$ a nonpositive function satisfying $$\Delta u + cu \geq 0$$. Prove that if $$u_{\mid \partial \Omega} \leq 0$$ then $$u_{\mid \Omega} \leq 0$$.

It is closely related to the weak maximum principle (WXP) ($$c=0$$ we get $$u$$ is a subharmonic function) but I do not know how to mimic the proof of the WXP.