 Solution

The divergence formula reduces the problem to the ordinary differential equation $r{f}^{\prime }+nf=0$ on $r>0$. Multiplication by $r^{n-1}$ makes it an exact derivative:Since the interval $r>0$ is connected, $r^{n}f(r)$ is a single constant $C$. Conversely, substituting $f=C{r}^{-n}$ into the divergence formula gives zero. ThusThis proves both existence and uniqueness up to the constant within the specified radial class. For $n=1$, the two punctured half-lines still share this same constant because the stipulated coefficient is one function of $r=|x|$.