Poisson equation with a spherical harmonic source ($\Delta (r^p{Y}_{\ell })=[p(p+1)-\ell (\ell +1)]r^{p-2}{Y}_{\ell }$)

Let $Y_{\ell }$ be a spherical harmonic of degree $\ell$ on the unit sphere. The Laplacian in spherical coordinates separates its angular part from a radial power, giving the displayed identity for $r>0$. Thus a Poisson equation $\Delta \Phi =r^q{Y}_{\ell }$ has the particular solution $\Phi =r^{q+2}{Y}_{\ell }/[(q+2)(q+3)-\ell (\ell +1)]$ when the denominator is nonzero. Add $A{r}^{\ell }{Y}_{\ell }+B{r}^{-\ell -1}{Y}_{\ell }$ to impose the radial boundary conditions. If the denominator vanishes at $p=q+2$, differentiating the displayed identity with respect to $p$ gives instead $\Delta (r^p\log r{Y}_{\ell })=(2p+1)r^{p-2}{Y}_{\ell }$. Regularity at the origin and decay at infinity must be checked separately; the radial identity alone is local away from zero.