I am trying to construct explicit solutions for the following PDE:
$$\Delta u-x\cdot \nabla u = 0$$
for $u:\mathbb{R}^3\to \mathbb{R}.$ In particular I am trying the following ansatz, $u(x)=A(r)B(\theta,\phi)$ where $(r,\theta,\phi)$ are the spherical co-ordinates. Then I know that,
$\Delta u = B\left(A_{rr}+\frac{n-1}{r}A_r\right)+\frac{A}{r^2}\Delta_{\mathbb{S}^{2}}B,$
where $u_r = \partial_r u.$ However, I am unsure how to write $x\cdot \nabla u$ in spherical coordinates. I changed the variable in the above PDE from cartesian to spherical coordinates and obtained that $x\cdot \nabla u = r u_r$, but I am not sure if my computation is correct.
So I would like to know if the PDE in spherical coordinates takes the form,
$$\left(u_{rr}+\frac{n-1}{r}u_r\right)+\frac{u}{r^2}\Delta_{\mathbb{S}^{2}}u-ru_r =0.$$