Consider $\Omega = \mathbb{R}^3 \setminus B(0,a)$ an exterior domain and $b \in \Omega$. I want to find the explicit solution of the following laplacian problem
\begin{aligned} - \Delta u = \alpha\delta(x-b) \text{ in } \Omega\\ u = 0 \text{ on } \partial \Omega\\ u(x) \underset{|x| \rightarrow +\infty}{\longrightarrow} 0 \end{aligned}
On the whole domain $\mathbb{R}^3$, $- \Delta u = \alpha\delta(x-b)$ would be easy to solve using the fundamental solution. However this is not possible here.
I'm looking for techniques or references to solve this problem. I was maybe thinking of something like boundary integral representation and layer potential techniques.
Any help or advice are welcomed !