Partial Derivatives & the Gradient · Superchallenge problem

Let $r = \sqrt{x^2 + y^2 + z^2}$. A function of the form $u(x,y,z) = g(r)$ satisfies $u_{xx} + u_{yy} + u_{zz} = 1$ in the shell $1 < r < 2$, and $u = 0$ on both spheres $r = 1$ and $r = 2$. Find the value of $u$ on the sphere $r = \dfrac{3}{2}$.