Solve

\begin{align}

&\Delta u=0 && x^2+y^2+z^2<1,\label{7-1}\\

&u=g(x,y,z) && x^2+y^2+z^2=1\label{7-2}

\end{align}

with $g(x,y,z)=z(x^2+y^2)$.

*Hint*. If $g$ is a polynomial of degree $m$ look for

\begin{equation}

u=g - P(x,y,z)(x^2+y^2+z^2-R^2)

\label{7-3}

\end{equation}

with $P$ a polynomial of degree $(m-2)$. Here $R$ is the radius of the ball. If $g$ has some rotational symmetry, so $P$ has.

**Bonus **

Represent $u$ as a sum of homogeneous harmonic polynomials.