Suppose $\Omega$ is a bounded smooth domain in $ R^N$ and $ a,b$ are bounded smooth vector fields in $ \Omega$. Suppose we have 
$$ -\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in } \Omega$$
$$-\Delta v(x) + b(x) \cdot \nabla u(x) = g(x) \ge 0 \quad \mbox{ in } \Omega$$ with $u=v=0$ on $ \partial \Omega$. I am curious whether there is a maximum principle for $u,v$ in the sense that $ u,v \ge 0$. I would prefer not to have any smallness or structural assumptions on $a,b$. 


I assume these are well known results (just not to me). 
Any comments are greatly appreciated.



Maximum principle for an elliptic system
Without any further assumptions, there is no such maximum principle. For example, in dimension $N = 1$, the functions $$u(x) = -1 + \cos x , \qquad v(x) = -\sin x$$ satisfy the system of elliptic equations given in the question with $f(x) = g(x) = 0$, $a(x) = 1$ and $b(x) = -1$, and both are equal to zero at the endpoints of $\Omega = (0, 2\pi)$. Nevertheless, neither $u$ nor $v$ is non-negative in $\Omega$.


