Solution
= Solution
The radial equation is
$$
(r^2\phi')'=-\frac1r.
$$
Its solutions satisfying decay at infinity have the form
$$
\phi(r)=\frac{\log r+1+C}{r}.
$$
The boundary value $\phi(1)=1$ gives $C=0$, so
$$
\boxed{\phi(r)=\frac{1+\log r}{r}}.
$$
It tends to zero as $r\to\infty$ and directly satisfies $\nabla^2\phi=-r^{-3}$.