I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind
$$
D\_i \left( a\_{ij} D\_j u \right) + b\_i D\_i u = 0.
$$
I assume the coefficients $a\_{ij},b\_i \in C^{\infty}(\mathbb R^n) \cap L^{\infty}(\mathbb{R}^n)$. 


Any hint/reference would be highly appreciated! 



A Liouville theorem for a uniformly elliptic equation in divergence form
What do you mean by the Liouville theorem? If the absence of bounded or positive harmonic functions, then the answer is "no" due to the presence of a vector field $b$. The corresponding counterexample can be constructed already for $n=1$. Take the diffusion coefficient $a=a\_{11}$ to be equal identically 1, and let $b=b\_1$ be odd and such that it converges to $+1$ at $+\infty$ (and therefore to $-1$ at $-\infty$). Then the space of bounded harmonic functions is 2-dimensional. This can be seen either directly, or from probabilistic considerations: the corresponding diffusion process on $\mathbb R$ converges to one of the two ends of the real line, so that it has a non-trivial behaviour at infinity.


