  We consider the stationary diffusion equation $-\mathrm{div} (\nabla u + bu
)=f$ in $n$-dimensional torus $\mathbb{T}^n$, where $f\in H^{-1}$ is a given
forcing and $b\in L^p$ is a divergence-free drift. Zhikov (Funkts. Anal.
Prilozhen., 2004) considered this equation in the case of a bounded, Lipschitz
domain $\Omega \subset \mathbb{R}^n$, and proved existence of solutions for
$b\in L^{2n/(n+2)}$, uniqueness for $b\in L^2$, and has provided a
point-singularity counterexample that shows nonuniqueness for $b\in L^{3/2-}$
and $n=3,4,5$. We apply a duality method and a DiPerna-Lions-type estimate to
show uniqueness of the solutions constructed by Zhikov for $b\in W^{1,1}$. We
use a Nash iteration to demonstrate sharpness of this result, and also show
that solutions in $H^1\cap L^{p/(p-1)}$ are flexible for $b\in L^p$, $p\in
[1,2(n-1)/(n+1))$; namely we show that the set of $b\in L^p$ for which
nonuniqueness in the class $H^1\cap L^{p/(p-1)}$ occurs is dense in the
divergence-free subspace of $L^p$.
