Consider $\Omega$ a smooth bounded domain in $\mathbb{R}^n$. Show that if $k=\dfrac{|\partial \Omega|}{|\Omega|}$ then the PDE $$\Delta u=k\ \text{in}\ \Omega,\ \dfrac{\partial u}{\partial n}=1\ \text{on}\ \partial \Omega$$ where $n$ is the normal to $\partial \Omega$, has a solution. Is the solution unique up to a constant?
When $\Omega$ is the unit ball, is there an explicit formula for $u$?
This is probably well-known stuff but I can't seem to find a reference, so if anyone could provide a proof or a reference it would be much appreciated.
That PDE with that boundary values has the weak formulation $$\int_{\Omega}\nabla u\cdot\nabla\varphi-k\varphi+\int_{\partial\Omega}\varphi=0$$ for every $\varphi\in C^{\infty}(\overline{\Omega})$. With this weak formulation, and with the trace inequality, it can be solved in $H^1(\Omega)$ in classical ways, for example with Lax-Milgram theorem.
If $\Omega$ is the unit ball, the solution is $\frac{1}{2}|x|^2$.