Laplace and Poisson equations

Let $B$ be the unit ball in $\mathbb{R}^3$. (a) Find the radial solution of $-\Delta u = 1$ in $B$, $u = 0$ on $\partial B$. (b) Explain why it is the only solution. (c) Use the maximum principle to show that any $v$ with $-\Delta v = f$, $|f|\le1$, $v = 0$ on $\partial B$ satisfies $|v|\le u$.