  We consider the Brezis--Nirenberg problem for the Laplacian with a singular
drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le
n \le 5$. The singular drift we consider derives from a potential which is
symmetric around the center of the (geodesic) ball. Here the potential is given
by a parameter ($\delta$ say) times the logarithm of the distance to the center
of the ball. In both cases we determine the exact region in the parameter space
for which positive smooth solutions of this problem exist and the exact region
for which there are no solutions. The parameter space is characterized by the
(geodesic) radius of the ball, $\delta$, and $\lambda$, the coupling constant
of the linear term of the Brezis-Nirenberg problem.
