I am interested in the existence of a global solution to the Poisson equation $$ \Delta\phi=g \quad \text{in} \ \mathbb{R}^n $$ such that $\phi\rightarrow 0$ when $|x|\rightarrow\infty$. My motivation for this comes from the solenoidal decomposition $$ f=f^{\text{s}}+\nabla\phi $$ where $f$ is a vector field on $\mathbb{R}^n$. I know that the solenoidal decomposition holds whenever $f$ has compact support ($f$ can be compactly supported distribution).
So my question is: what are the optimal assumptions for the existence and uniqueness of a classical solution for the solenoidal decomposition? I know that solenoidal decomposition exists for Schwartz vector fields, but I would like to have lower regularity. This can be reformulated in terms of the Poisson equation since the potential $\phi$ is defined so that it solves the equation $\Delta\phi=\text{div}f$. The uniqueness holds whenever $\phi$ goes to zero at infinity. But what regularity I have to assume for $f$ in order to have existence? Is differentiability and rapid decay enough? In the case of a bounded domain $\Omega$ I know that $\text{div}f\in C^{\alpha}(\overline{\Omega})$ for some $0<\alpha<1$ is enough to have unique solution for the boundary value problem \begin{align} \Delta\phi &=\text{div} f \quad \text{in} \ \Omega \\ \phi&=0 \quad \text{on} \ \partial\Omega \end{align}
References to books/articles are appreciated. I already checked Evans and Gilbarg & Trudinger but I did not find what I was looking for. Thanks in advance.