$\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ has at most one solution

Question

I want to show that the PDE $\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ with the boundary value $u = \varphi$ on $\partial \Omega$ has at most one solution under the following conditions:

1. $\Omega$ is a bounded Domain
2. $f$ is continuous in $\mathbb{R} \times \overline{\Omega}$ and continously differentiable in u so that $f_u(u,x) \ge 0$

I believe the maximum principle might be applicable here, but I'm unsure about how to prove that $\Delta (u-v) \ge 0$ for 2 solutions of the equation

If anyone could provide a hint on how to solve this, I would greatly appreciate it.