= Solution
Write $R=|x-x_0|$. A radial function <harmonic> away from the source satisfies $(R^2G')'=0$, hence $G=A+B/R$. Decay at infinity forces $A=0$. Integrating $\Delta G=\delta(x-x_0)$ over a small sphere, the <divergence theorem> requires flux one:
$$
4\pi\epsilon^2G'(\epsilon)=-4\pi B=1.
$$
Therefore the <fundamental solution> in this sign convention is
$$
\boxed{G_0(x,x_0)=-\frac1{4\pi|x-x_0|}.}
$$
Adding point sources outside the domain changes only a <harmonic function> inside it. The <method of images> selects their positions and strengths to cancel boundary values. The sign reverses for the alternative convention $-\Delta G=\delta$.
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