Consider (*): $\Delta u=f$ in $\mathbb{R}^N$.
Assume that $f \in L^2(\mathbb{R}^N)$ is such that $f=0$ outside a compact. Is it true that (*) has a unique solution in $H^2(\mathbb{R}^N)$ ? How to prove it ?
More generally, I am looking for references on this equation in $\mathbb{R}^N$ for weak solutions (existence, uniqueness, regularity, etc.). The Evans only works with $C^2$ solutions it seems.