I would like to find an explicit example of a linear elliptic operator having the following form:
$$Lu=-\Delta u +b(x)\cdot \nabla u, $$
where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that there exists a non trivial (weak) solution of the Dirichlet problem
$$ \begin{cases} Lu = 0 & \text{in}\ \Omega \\ u=0 & \text{on}\ \partial \Omega\end{cases}.$$
Here $\Omega$ can be any bounded and smooth domain of $\mathbb{R}^n$. 

In functional terms, I am trying to convince myself that a first-order perturbation of the Laplacian can alter the spectrum to the point that $0$ becomes an eigenvalue. 

Can somebody provide me with such an example? I have tried looking at the one-dimensional case, when $\Omega$ reduces to an interval, but I could not find one.

Thank you for reading.