 Solution

For two solutions let $u$ be their difference. Green’s identity and ${\partial }_{n}u=0$ give $0={\int }_{V}u({\nabla }^{2}u-m^{2}u)=-{\int }_V(|\nabla u{|}^2+m^2{u}^2)$, so $u=0$ when $m\ne 0$. At $m=0$, solutions are unique only up to an additive constant.