Suppose we want to solve Poissons equation$$\nabla^{2}\phi=f$$ for a point source at the origin so $$f=k\delta(\textbf{r}')$$ then the solution is given by $$\phi(\textbf{r})=-\int\frac{k\delta(\textbf{r}')}{4\pi|\textbf{r}-\textbf{r}'|}d^{3}r'=-\frac{k}{4\pi|\textbf{r}|}$$ I now want to add a volume, $\alpha$, defined by the boundary $\partial\alpha$ that does not overlap with the origin. In this volume I would like $\phi(\textbf{r})$ takes some fixed value $\phi_{0}$, that is $$\phi(\textbf{r})=\begin{cases}\phi_{0}\;\text{if}\;\textbf{r}\;\text{is in}\;\alpha\\ \Phi(\textbf{r})\;\text{if}\;\textbf{r}\;\text{is outside}\;\alpha \end{cases}$$ how do I find $\Phi(\textbf{r})$? I honestly have no clue where to begin so any help is greatly appreciated.
Edit: The context of this problem is that I am trying to resolve the electric potential field of a system consisting of a point charge and a conductor held at a fixed potential.
I doubt it's possible to obtain an analytic solution, what I'm looking for is some guidance of how I could set this problem up to be solved numerically. All I know is that $$\nabla^{2}\phi=k\delta(\textbf{r})$$ and that we have the boundary conditions that $\phi=\phi_{0}$ on $\partial\alpha$ and $\phi=0$ as any element of $\textbf{r}$ approaches $\pm\infty$.
The problem as stated is missing a boundary condition on $\phi$ as $\|x\| \to \infty$. From the context (electrostatics), I assume that the missing condition is $\phi(x) \to 0 \quad \text{as } \|x\| \to \infty$.
I would pose the problem as follows: Let $\Omega = \mathbb{R}^3\setminus \alpha$ be the domain. Note that $\partial \Omega = \partial \alpha$. We would like to solve the boundary value problem $$ \nabla^2 \phi = f \quad \text{on } \Omega,\\ \phi(x) = \phi_0 \quad \text{for } x \in \partial \Omega,\\ \phi(x) \to 0 \quad \text{as } \|x\| \to \infty,\tag{1}\label{eq1} $$ where $f = k \delta$. Since the equation is linear, we can use the following approach: Compute that solution $\phi_1$ to the boundary value problem $$ \nabla^2 \phi_1 = 0 \quad \text{on } \Omega\\ \phi_1(x) = \phi_0 - k \Phi(x) \quad \text{for } x \in \partial \Omega\\ \phi_1(x) \to 0 \quad \text{as } \|x\| \to \infty,\tag{2}\label{eq2} $$ where $\Phi(x) = -\frac{1}{4 \pi x}$ is the fundamental solution. Then we notice that the function $\phi = \phi_1 + k \Phi$ satisfies $$ \nabla^2 \phi = \nabla^2 \phi_1 + \nabla^2 k \Phi = 0 + f = f. $$ Moreover, it satisfies both boundary conditions by construction.
Now the problem has been reduced to solving a boundary value problem for Laplace's equation on the (unbounded) domain $\Omega$. The solution certainly exists as long as $\partial \Omega$ is sufficiently regular (Lipschitz), but for most $\alpha$, you may have to resort to numerics.