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 $\Omega\subset \Bbb R^2$containing the origin, where $f\in W^{-1}_q(\Omega)$ with $q > 2$ and $b(\alpha):=b-\frac{\alpha x}{|x|^2}$ , $b$ is a divergence-free vector field and $\alpha\in\Bbb R$ is a parameter. Then we further improved this result to three dimensional case with a bounded domain $\Omega \subset\Bbb R^3$ and drift $b(\alpha):=b-\frac{\alpha x'}{|x'|^2}$, where $x'=(x_1,x_2,0)$.