Potential of a uniform spherical source ($\Delta f={\rho }_0{1}_{\{|x|<a\}}$)

For a uniform spherical source of radius $a$, the origin-regular radial Poisson equation gives $\nabla f={\rho }_{0}x/3$ inside and $\nabla f={\rho }_0{a}^{3}x/(3|x{|}^3)$ outside. Integrating radially, imposing $f\to 0$ at infinity and continuity at $|x|=a$, givesThe potential and its normal derivative are continuous at the interface, so its distributional Laplacian has no surface delta contribution. The classical equation holds off the interface and the source equation holds almost everywhere and distributionally.