Presentation of the problem :
We have a uniform homogenous isotropic dielectric sphere in an electrostatic field.
To solve this problem, we remark that we have an azimuthal symmetry. So the potential of the problem is $V(r, \theta)$.
Because we are in homogenous isotropic dielectric medium, we have a Laplace equation for the potential inside and outside of this sphere:
$$ \Delta V=0$$
By assuming $V=f(r)g(\theta)$, we end up by a potential of the following form:
$$ V(r,\theta)=\sum_{l=0}^{\infty} (A_l r^l+B_l r^{-(l+1)} )P_l (\cos(\theta)) $$
This solution is done in Plasmonic fundamentals by Stefan Alexander Maier. The image is taken from it.
My problem :
The moment I don't totally understand is that they say we decompose the potential in two contributions: $V_{in}$ inside the sphere and $V_{out}$ outside.
Then they say that as the potential can't diverge in $r=0$, we have:
$$ V_{in}=\sum_{l=0}^{\infty} (A_l r^l)P_l (\cos(\theta)) $$
And
$$ V_{out}=\sum_{l=0}^{\infty} (C_l r^l+D_l r^{-(l+1)})P_l (\cos(\theta)) $$
What I don't understand is: how do we know we have to write two contributions of the potential.
If I had to solve the problem by myself I would only write one potential $V$ as $$ V(r,\theta)=\sum_{l=0}^{\infty} (A_l r^l+B_l r^{-(l+1)} )P_l (\cos(\theta)) $$
And then saying that the potential can't diverge in $r=0$, thus I would write $A_l=0$.
But of course it would be wrong (not enough "unknown to solve").
But I don't see from a math point of view why I should write two different potentials. Indeed, the potential $V$ is following the same equation $\Delta V=0$ inside and outside of the sphere so how can I know I have to write it in two different ways $V_{in}$ and $V_{out}$ ?