Maximum principle for an elliptic system

Question

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.

Answer 1

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 $\Omega$.