  We examine the equation given by \begin{equation} \label{eq_abstract} -\Delta
u + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where
$p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We
show that for $ p < p_c$, the Joseph-Lundgren exponent, that there is no
positive stable solution of (\ref{eq_abstract}) provided one imposes a
smallness condition on $a$ along with a divergence free condition. In the other
direction we show that for $ N \ge 4$ and $ p > \frac{N-1}{N-3}$ there exists a
positive solution of (\ref{eq_abstract}) provided $a$ satisfies a smallness
condition. For $ p>p_c$ we show the existence of a positive stable solution of
(\ref{eq_abstract}) provided $a$ is divergence free and satisfies a smallness
condition.
