I would like to calculate $\Delta_r f(r) $. This is as far as I got:
$\Delta_r f(r) =\nabla \cdot \nabla f(r) =\nabla \cdot \frac{\partial }{\partial r} f(r) \hat{r} = \nabla ( \frac{\partial }{\partial r} f(r) ) \cdot \hat{r} + ( \frac{\partial }{\partial r} f(r) ) \nabla \cdot \hat{r} $
$=\frac{\partial^2 }{\partial r^2 } f(r) \hat{r}\cdot \hat{r}+( \frac{\partial }{\partial r} f(r) ) \nabla \cdot \hat{r}$
$=\frac{\partial^2 }{\partial r^2 } f(r) +( \frac{\partial f}{\partial r} ) \nabla \cdot \hat{r}$
Now how can I proceed? In other words: how can I calculate $\nabla \cdot \hat{r}$?
$$\nabla \cdot r = \sum_{i=1}^n \partial_i \bigg(\frac{x_i}{\sqrt{x_1^2 + \cdots + x_n^2}}\bigg) = \frac{n}{r} - \sum_{i=1}^n \frac{x_i^2}{(x_1^2 + \cdots + x_n^2)^{3/2}} = \frac{n}{r} - \frac{1}{r} = \frac{n-1}{r}\ .$$