Laplacian of powers of the norm in R^n

Question

Consider the function $f(x) = \lvert x\rvert^{2-n}$, $x \in \mathbb{R}^n$ \ $\{0\}$.

I have derived the following expression for the Laplacian

$$(2-n) n \lvert x\rvert^{-n} - \frac{(2-n)n}{2} \sum^n_{j=1} x_j^{-n/2}$$

However I am not convinced of the correctness of my answer. Could someone confirm if this is the right result?

Thanks in advance.

Answer 1

Let $m = 2-n$ then I got the following

$$\triangle \|x\|^m = m(m-2)\|x\|^{m-2}+ m \|x\|^{m-2}\sum_{j=1}^n x_j$$