I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving
$$
\left\{
\begin{align}
& \Delta \psi = - \rho & & \mbox{in } \mathbb{R}^3, &(1)
\\
& \psi(\infty) = 0. & & &(2)
\end{align}
\right.
$$
where $\rho$ is a compactly supported function.

We know the answer should be given (up to constant factors) by
\begin{equation}
\psi(x) = \int_{\mathbb{R}^3} \frac{\rho(y)dy}{|x-y|}. \qquad (3)
\end{equation}
My first question is what are the mildest conditions that can be imposed on $\rho$ for (1) and (2) to hold pointwise, and is the solution indeed given by (3)? I think Gilbarg and Trudinger give it to hold for $\rho$ Holder continuous with Holder exponent $\alpha \in (0,1]$.

My second question is what if I relax the condition that (1) hold pointwise, and instead seek weak solutions i.e. $\psi$ satisfying $$ \int_{\mathbb{R}^3}\left[ \nabla\psi\cdot\nabla\varphi - \rho\varphi\right]dx = 0, \qquad \psi(\infty)=0 \qquad (4) $$ should hold for all test functions (i.e. $C_c^\infty$) $\varphi$? Then how does the generality improve? Can it be broadened to allow for measures $\rho$ if we replace $\int\varphi\rho dx$ by $\int \varphi d\rho$ in (4)?

FYI the books I have been consulting include Evan's PDE, Gilbarg and Trudinger's Elliptic PDE, Landkof's Potential Theory, Helms' Potential Theory, Jackson's Electrodynamics. Thank you all in advance.