 Solution

For any rotation $R$ about the centre, ${\varphi }_R(x)=\varphi (Rx)$ satisfies the same equation and the same Dirichlet data because $V,f,g$ are rotationally invariant. Uniqueness from part (a) gives ${\varphi }_R=\varphi$ for every $R$. Thus $\varphi$ is constant on spheres and is a function of $r$ alone.