Look for a solution in the form $u=P(\rho)\Phi(\phi)\Theta(\theta)$.
Write equations and their solutions for $P$ and $\Theta$.
Write equation (with parameters $l$ and $m$) for $\Phi$; what values can take $(l,m)$?
Problem 2.
For equation to $\Phi$ find solutions in the form $L(\cos(\phi),\sin(\phi))$ where $L$ is a polynomial of degree $l$ with respect to $\cos(\phi), \sin(\phi)$ in the following cases:
$l=0$ and all possible values $m$;
$l=1$ and all possible values $m$;
$l=2$ and all possible values $m$;
$l=3$ and $m=0$.
Problem 3.
Solve
\begin{align}
&\Delta u=0 && x^2+y^2+z^2< a^2,\label{8.P.1}\\
&u=g(x,y,z) && x^2+y^2+z^2=a^2 \label{8.P.2}
\end{align}
with $g(x,y,z)$ defined below and $a=1$. 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-a^2)
\label{8.P.3}
\end{equation}
with $P$ a polynomial of degree $(m-2)$. Here $a$ is the radius of the ball. If $g$ has some rotational or reflectional symmetry, so $P$ has.
Represent $u$ as a sum of homogeneous harmonic polynomials.
Problem 4.
Find function $u$, harmonic in $\{x^2+y^2+z^2\ge a^2\}$, decaying at infinity, and coinciding with $g(x,y,z)$ as $x^2+y^2+z^2=a^2$ with $g(x,y,z)$, defined in Problem 3 (a)-(j), and $a=1$.
Hint. Combine Problem 4 and Problem 7.P.6.