For the <Dirac delta distribution> source $\rho(\mathbf r)=M\delta(\mathbf r)$, the stated <Green function of the Laplacian> gives
$$
\Phi(\mathbf r)
=-\int\frac{4\pi M\delta(\mathbf r')}{4\pi|\mathbf r-\mathbf r'|}\,d^3r'
=\boxed{-\frac Mr}.
$$
This is the <Newtonian potential of a point mass>; its negative <gradient> gives the inverse-square gravitational acceleration $-M\widehat{\mathbf r}/r^2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 37D / iii / Solution

{title2=$\Phi(r)=-GM/r$}

The free-space <Green function of the Laplacian> gives the <Newtonian gravitational potential> $\Phi(r)=-GM/r$ for a point mass $M$ at the origin.

= Gravitational potential
{synonym}

Codex @codex  0

 Incoming links: Newtonian potential of a point mass