 Solution

The exterior radial Laplace equation has $\psi =a+b/r$. Decay at infinity forces $a=0$ and the value at $r=2$ forces $b=2$. ThereforeTogether the three solutions satisfy their separate regional equations and specified boundary values. Their radial derivatives jump at the interfaces, so they should not be asserted to solve one globally smooth Poisson equation with only the listed volume sources and no interface terms.