  We consider nonnegative solutions to $-\Delta u=f(u)$ in unbounded euclidean
domains, where $f$ is merely locally Lipschitz continuous and satisfies
$f(0)<0$. In the half-plane, and without any other assumption on $u$, we prove
that $u$ is either one-dimensional and periodic or positive and strictly
monotone increasing in the direction orthogonal to the boundary. Analogous
results are obtained if the domain is a strip. As a consequence of our main
results, we answer affirmatively to a conjecture and to an open question posed
by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and
monotonicity results in the higher-dimensional case.
