The Laplace operator. What will be the value of $\Delta\left(\frac 1{r^2}\right)$ if $r=|x|=\sqrt{x_1^2+x_2^2+x_3^2}$?
May be this can be determined using Green's formula.

This is not a PDE. This is just the Laplacian. A PDE is an equation. See https://en.wikipedia.org/wiki/Laplace_operator#Coordinate_expressions.
$$
\Delta\left[\frac{1}{r^{2}}\right]=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left[r^{2}\frac{\partial}{\partial r}\left[\frac{1}{r^{2}}\right]\right]=\frac{2}{r^{4}}
$$
