Attempt
In my book, we have the definition for Green's Function
$$ G(r',r) = \frac{1}{4 \pi } \frac{1}{|r'-r|} + h(r,r')$$
is the Green's function for the Dirichlet problem on some domain of $R^3$ and $h$ is to satisfy the following: $\Delta h(r') = 0$ and $h(r') = - \frac{1}{4 \pi } \frac{1}{|r'-r|}$ for $r'$ in the boundary of the domain. But, we want to satisfy $\Delta G = \delta (x-x_0)$. How do we apply this condition to find our desired Green's Function? Im quite confused how to derive such expressions