I am interested in solving the following PDE,
$$\Delta u + w(|x|) x\cdot \nabla u = 0,$$
where $u:\mathbb{R}^3 \to \mathbb{R}.$
I know that, $$\Delta u = u_{rr} + \frac{n-1}{r}u_r + \frac{1}{r^2}\Delta_{S^2}u$$
and after computation I deduced that $x\cdot \nabla u = r u_r.$
Therefore I am wondering if a general solution to this PDE is of the form, $$u(r,\theta,\phi) =\sum_{l}\sum_{m=-l}^{l}a_{ml}g_l(r)Y_{ml}(\theta,\phi)$$
where $g_l(r)$ solves, $$g''+\left(\frac{n-1}{r}+rw(r)\right)g'-\frac{l(l+1)}{r^2}g=0$$
and the spherical harmonics solve, $$-\Delta_{S^2}Y = l(l+1)Y.$$
Is this expansion correct?