TITLE: Looking for a first order perturbation of the Laplacian having 0 in its spectrum
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: I am starting to think that such an example is not that easy to find. Indeed, it won't exist if $b$ is the gradient of a scalar field $B$. I propose a proof of this: please, let me know if you find it wrong. Thank you.
Claim Suppose that $B\colon \mathbb{R}^n\to \mathbb{R}$ is differentiable. Then the only weak solution to the problem
$$\tag{1}\begin{cases} -\Delta u + \nabla B\cdot \nabla u =0 & \Omega \\ u=0 & \partial \Omega\end{cases}$$ 
is the trivial one. 
Proof As you can see in this answer, the problem (1) is the Euler-Lagrange equation of the following functional:
$$S(u)=\int_{\Omega} e^{-B(x)}\frac{\lvert \nabla u\rvert^2}{2}\, dx,$$
so that (1) is equivalent to the relation 
$$dS(u)v=0, \quad \forall v \in H^1_0(\Omega).$$
We compute $dS(u)v$ and discover that 
$$dS(u)v=\int_\Omega e^{-B(x)}\nabla u\cdot \nabla v\, dx.$$
So if $u$ satisfies (1) we have, setting $v=u$ above, 
$$\int_\Omega \lvert \nabla u\rvert^2e^{-B(x)}\, dx=0,$$
from which we infer that $\nabla u =0 $ on $\Omega$.