Problem in English (original problem 7 on page 813 here)
Suppose $f(|\bar{x}|)=\sqrt{x_{1}^2+...+x_{n}^{2}}$. For what kind of real $f$ it holds that $f$ is harmonic everywhere but not in origin? If $f$ is harmonic, then $\triangle f=0$.
Definitions
The "real function" apparently here means some $g$ such that $g: \mathbb R^{n}\to\mathbb R$, not vector in the co-domain but scalar (can be realized by looking at the norm) but $\bar{x}\in\mathbb R^n$ (please verify).
I am not sure whether this problem is just a brute-force calculation -practise or some clever trick, below some of my calculations for one term, not summing it up because it is a messy.