Let $f\in C^1(\mathbb{R}^3)\cap L^1 (\mathbb{R}^3)$, then there exists a solution $u\in C^2(\mathbb{R}^3)$ solving $-\Delta u = f$ on $\mathbb{R}^3$ given via $$u(x) = \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{f(y)}{|x-y|}\ dy$$.
N.B I know the proof of this theorem when $f\in C^2_c(\mathbb{R}^3)$ as given in Evans. For this data I tried to use a cutoff function on f and a density argument but I don't feel I'm right.
This statement is given in Majda-Bertozzi book on Vorticity and Incompressible flow (page 38) without proof.
Can anyone point me towards a reference where I can get a proof?
The key piece(s) in the Evans proof are integration by parts and the use of the compact support of $f$, right?
I believe that the assumption that $f$ is continuous with one cts derivative and that $f$ is in $L^1$ are enough to proceed similarly. In places where before you have vanishing or decaying terms because of the compact support of $f$, now you should have similar behavior due to $f \in L^1$. Recall that if $f$ is in $L^1$ you can say that $f$ is less than any fixed number outside a set of finite measure - for otherwise $f$ is at least some number $a$ on a set of infinite measure and therefore isn't in $L^1$.
Using this, I believe you can pick a small number $\epsilon$, place balls around the sets where $f$ is at least $\epsilon$ and use arguments similar to those in the Evans proof, and outside these balls use that $f <\epsilon$ to control the behavior here.