I want to show that the solution $u\in \mathcal{C}^2(\Omega)\cap\mathcal{C}^0(\bar{\Omega})$ to the following Poisson boundary value problem $$ \begin{cases} r=-\Delta u & \textrm{in } \Omega\\ 0=u & \textrm{on } \partial\Omega \end{cases} $$ where $\Omega\subset B_n(0,R)$ the $n$-dimensional ball of ray $R>0$, and $r\in\mathcal{C}^0(\bar{\Omega})$, is bounded according to the following inequality: $$ \min_{y\in\bar{\Omega}}(0,r(y))\frac{R^2-|x|^2}{2n}\leq u(x)\leq \max_{y\in\bar{\Omega}}(0,r(y))\frac{R^2-|x|^2}{2n} $$
It must be a link somewhere with the Poisson formula, but I don't see how..
All help welcome !