I am curious whether there are any Liouville theorems for the following pde:

$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$

where $a(x)$ is a smooth bounded vector field with $ |x| |a(x)| \rightarrow 0$ as $ |x| \rightarrow \infty$. As far as the solutions $ \phi$ I am assuming that $ \phi$ is smooth and the gradient satisfies a decay condition like $ | \nabla \phi(x)| \le C |x|^{-\sigma}$ for large $ x$ where $ \sigma>0$ is some small parameter.

As a second question, how about if we additionally assume $a(x)$ is divergence free.

Note if $ \phi$ went to zero at $\infty$ then we can get a Liouville theorem directly from the maximum principle. So one think I tried is taking a derivative of the equation to get a solution which decays... but I am not having much luck.

any comments or counterexamples would be welcomed. Craig