Let $u$ be a $C^{2}$ harmonic function in $\mathbb{R}^{n}$ and let $g(x) = \left| x \right|^{2-n}$.
I would like to show that:
$v(x) = g(x)u\left(\frac{x}{\left| x \right|^{2}}\right)$
is also harmonic where it is defined, and I am wondering if there is a way to do this without explicitly showing that the Laplacian is zero. I believe that g(x) is also harmonic and radial, but I also know that the product of two harmonic functions is not necessarily harmonic.
Maybe there is a clever way to solve this problme, but I don't know it. The usual solution is just use the definition which implies in calculating $\Delta v(x)$. You can try this boring calculation, or you can take a look here where the calculation is made.