Is there a non brute force way to show $v(x)=|x|^{2-n}u\left(\frac{x}{|x|^2}\right)$ is harmonic if $u$ is harmonic on a domain of $\mathbb{R}^n$?
I have tested this for the fundamental solution and verified the mean value over spheres centred at the origin. The mean value property over spheres not entered at the origin seems pretty brute force like computing the laplacian. Wondering if there is a better way.
I also tried to use the product rule $\Delta (fg) $ but did not get so far because $\Delta_x u\left(\frac{x}{|x|^2}\right)$ is kind of complicated.
I tried to interoperate what $v$ means. The map $x\mapsto \frac{x}{|x|^2}$ is the inversion about the unit sphere, and so the claim appears to be saying if we adjust the inversion up to a factor of $|x|^{2-n}$ (which is also harmonic) then the function $v$ obtained is also harmonic. How could this observation help?