Stability of entire solutions to supercritical elliptic problems involving advection

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.