Scalar elliptic equations with a singular drift

Abstract

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the origin, where $f \in W^{-1}_q(\Omega) $ with $q>2$ and $b^{(\alpha)}:=b-\alpha \frac{x}{|x|^2}$, $b$ is a divergence-free vector field and $\alpha\in {\mathbb R}$ is a parameter.