Poisson potential of an exponential-over-radius source ($\rho (r)=A{e}^{-\beta r}/r$)

For $\beta >0$, solve ${\nabla }^2\varphi =-\rho$ in three dimensions with $\rho (r)=A{e}^{-\beta r}/r$, finite potential at the origin and decay at infinity. The Green function is $1/(4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|)$ in this sign convention. The angular integral supplied by the spherical shell theorem givesThe value at the origin is the limit $A/\beta$. A second derivation sets $u=r\varphi$: the radial Poisson equation becomes $u^{\prime \prime }=-A{e}^{-\beta r}$. Decay excludes the constant-potential term and boundedness at the origin cancels the remaining $1/r$ singularity. Although the source is locally integrable, it is singular at the origin; the resulting potential solves the equation there in the sense of distributions, without an extra Dirac delta.