 Solution

Translation and rotation invariance allow a radial Green function $G(R)$, with $R=|\mathbf{r}-{\mathbf{r}}^{\prime }|$. Outside the source, $G^{\prime \prime }+2G^{\prime }/R=0$, so $G=A/R+B$. Decay at infinity gives $B=0$.

Integrate the defining equation over a small ball centred on the source. The divergence theorem fixes its flux:ConsequentlyThis is the Green function of the Laplacian with the stated negative-delta sign. A homogeneous $1/R$ addition would change the delta normalization, so the singular coefficient is fixed, not an arbitrary integration constant.