Given a grounded conducting sphere, $V=0$ and $\text{radius} = R$, centered at the origin with a pure electric dipole (dipole moment $\vec p$) situated at the origin and pointing along the positive $z$ axis, I should be able to solve Laplace's equation in spherical coordinates to find the potential everywhere in the sphere. I can separate the differential equations and use Legendre polynomials, but I'm having trouble defining and using my boundary conditions. What I (think I) know so far:
$$V(r,\theta)=\sum_{n=0}^\infty (A_n r^n + \frac{B_n}{r^{n+1}}) P_n(\cos{\theta})$$
$$r\to R \Rightarrow V\to 0$$ $$\theta \to \frac{\pi}{2} \Rightarrow V\to 0$$ $$\theta \to 0 \space or \space \pi \Rightarrow V\to \frac{\hat{r}\cdot \vec{p}}{4\pi\varepsilon_0r^2}$$
At least I think those boundary conditions work. I need to define a condition as well for the case of $$r \to 0$$ but that would seem to make the potential explode. How can I use these boundary conditions to solve for the potential everywhere inside the sphere?