Shell theorem for a general potential

The Yukawa potential $u$ in empty space satisfies $\Delta u = a^2 u$. There are two linearly indpendent spherically symmetric solutions of this differential equation: $\exp(-ar)/r$ and $\exp(+ar)/r$. The potential outside the spherical shell should be spherically symmetric and decay, not grow, as $r \to \infty$, so this must be $C \exp(-ar)/r$ for some constant $C$. This is the potential of a point mass $C$ at the origin. On the other hand, inside the shell the potential should be nonsingular at the origin, so it is of the form $A \sinh(ar)/r$, which has a finite limit $Aa$ as $r \to 0+$. For the potential to be continuous at the shell itself (radius $R$), we need $A \sinh(aR)/R = C \exp(-aR)/R$, i.e. $C = A \sinh(aR) \exp(aR)$.

Now if the shell has radius $R$, the origin is at distance $R$ from all points of the shell, so the potential at the origin is the same as it would be if all the mass of the shell was gathered into one point, still at distance $R$, namely $M \exp(-aR)/R$ where $M$ is the mass of the shell. Thus $Aa = M \exp(-aR)/R$. Thus we get $C = M \sinh(aR)/(aR)$.