Find a bounded but non-constant solution of Poisson's equation

Question

Consider Poisson's equation with homogenous Dirichlet boundary conditions \begin{equation} \begin{split} \Delta u&=g \text{ in } B_R(0)\\ u&=0 \text{ on } \partial B_R(0), \end{split} \end{equation} where $B_R(0) \subset \mathbb{R}^d$ is a sphere around the origin with radius $R>0$. Can we find an explicit example of a solution $u$ (and a corresponding function $g$) of this problem that is bounded but not constant? Is this even possible?

Answer 1

Sure. Just take any non-constant function $u$ that satisfies the boundary conditions and let $g=\Delta u$. If you want it to be explicit, you can take $u(x)=|x|^2-R^2$ and $g=2d$.