In this paper we study existence, uniqueness, and integrability of solutions
to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are
particularly interested in singular E with divE≥0. We start
by recalling known existence results when ∣E∣∈LN that do not rely on the
sign of divE. Then, under the assumption that divE≥0 distributionally, we extend the existence theory to ∣E∣∈L2. 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 divE∼dist(x,∂Ω)−2−α. 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. ∂u/∂n<0, fails in the presence of such
singular drift terms E