 Potential of an inverse-radius spherical charge density

A charge cloud $\rho =A/r$ for $0<r<a$ encloses $Q(r)=2\pi A{r}^2$. With no central point charge and potential zero at infinity, Gauss's law gives $\varphi =A(2a-r)/(2{ϵ}_0)$ inside and $A{a}^2/(2{ϵ}_{0}r)$ outside. The density singularity is integrable, and the potential is finite at the origin.