The Chain Rule & Implicit Differentiation · Superchallenge problem

A smooth function on all of $\mathbb R^3$ has the radial form $u = g(r)$, where $r = \sqrt{x^2 + y^2 + z^2}$, and satisfies $u_{xx} + u_{yy} + u_{zz} = u$ everywhere (including the origin), with $g(0) = 1$. Find $g(\ln 2)$.