  We study the solutions $u$ to the equation $$ \begin{cases}
\operatorname{div} u + \langle a , u \rangle = f & \textrm{ in } \Omega,\\ u=0
& \textrm{ on } \partial \Omega, \end{cases} $$ where $a$ and $f$ are given. We
significantly improve the existence results of [Csat\'o and Dacorogna, A
Dirichlet problem involving the divergence operator, \textit{Ann. Inst. H.
Poincar\'e Anal. Non Lin\'eaire}, 33 (2016), 829--848], where this equation has
been considered for the first time. In particular, we prove the existence of a
solution under essentially sharp regularity assumptions on the coefficients.
The condition that we require on the vector field $a$ is necessary and
sufficient. Finally, our results cover the whole scales of Sobolev and H\"older
spaces.
