Pointwise estimate for solutions of the laplace equation on bounded domains

Let $v=u+\frac{|x|^2}{2n}\lambda$, where $\lambda=\max_{x\in \overline{\Omega}}$. Then
$$-\Delta v=-\Delta u-\lambda=f-\max_{x\in \overline{\Omega}} |f| \leq 0.$$
Hence $v$ is submharmonic, and by the weak maximum principle for subharmonic functions we have $$\max_{x\in \overline{\Omega}} u \leq \max_{x\in \overline{\Omega}} v =\max_{x\in \partial \Omega} v \leq \max_{x\in \partial \Omega } (u+ \frac{|x|^2}{2n}\lambda) \leq \max_{x \in \partial \Omega}|g|+C_1 \lambda.$$ 
Hence $$\max_{x\in \overline{\Omega}} u \leq C_2( \max_{x \in \partial \Omega}|g|+ \lambda)=C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|),$$ for some constant $C_1, C_2$ which only depend on $\Omega$. By replacing $u$ with $-u$, we get
$$\max_{x\in \overline{\Omega}} (-u) \leq C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|).$$ 
Combining the above two inequalities we get $$\max_{x\in \overline{\Omega}} |u| \leq C( \max_{x \in \partial \Omega}|g|+ \max_{x\in \overline{\Omega}} |f|).$$