Summary: In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem \(-\mathrm{div}(M(x) \nabla u) = -\mathrm{div} (E(x) u) + f\) in a bounded domain of \(\mathbb{R}^N\) with \(N \geq 3\). We are particularly interested in singular \(E\) with \(\mathrm{div}\, E \geq 0\). We start by recalling known existence results when \(|E| \in L^N\) that do not rely on the sign of \(\mathrm{div}\, E\). Then, under the assumption that \(\mathrm{div}\, E \geq 0\) distributionally, we extend the existence theory to \(|E| \in L^2\). For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of \(E\) singular at one point as \(Ax /|x|^2\), or towards the boundary as \(\mathrm{div}\, E \sim \mathrm{dist}(x, \partial \Omega)^{-2-\alpha}\). In these cases the singularity of \(E\) leads to \(u\) vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. \(\partial u/\partial n < 0\), fails in the presence of such singular drift terms \(E\).