Let $U \subseteq \mathbb R^3$ be a star shaped and $F: U \to \mathbb R^3$ a continuably differentiable vector field. How can I now show that $F$ can be written as a sum $$F=G+H$$ where $$rot \, G=0 \, , \, div \, H =0$$ if and only if the differential equation $\Delta u=div \, F$ has a solution $u:U \to \mathbb R$.
A little help or a hint is much appreciated.
$\def\rot{\operatorname{rot}} \def\div{\operatorname{div}}$
If $u$ solves $\Delta u =\div F$, then choosing $G = \nabla u$ and thus $H = F-\nabla u$ gives $\rot G = \rot \nabla u =0$ and $\div H = \div F - \div \nabla u = \Delta u - \div \nabla u = 0$.
Conversely, suppose that $F = G + H$ with $\rot G = \div H = 0$. Then by the Poincaré Lemma we can find a potential $u : U \to \mathbb R$ such that $\nabla u =G$, and we can calculate
$$ \Delta u = \div \nabla u = \div G = \div F - \div H = \div F$$ as desired.