I have a question that I'm getting stuck on. I am given $R>0$, and $c,f\in C(\overline{B_R})$, where $c \leq 0$ on $\overline{B_R}$ and assume that $u \in C^2(B_R) \cap C( \overline{B_R})$ satisfies $$ \begin{cases} \Delta u + cu = f & x \in B_R \\ u = 0 & on\,\,\, \partial B_R \end{cases} $$ Prove that $$ \sup_{B_R} |u| \leq \frac{R^2}{2n} \sup_{B_R} |f| \,\,. $$ Hint: let $A =\sup_{B_R} |f|$ and define $v(x)= \frac{A}{2n} (R^2 - |x|^2)$ and prove that $|u(x) \leq v(x)$ on $B_R$.
I'm thinking I need to use a maximum principle type argument to show this result, but I'm getting stuck on how to start this.