Monotonicity for solutions to semilinear problems in epigraphs and applications

We consider positive solutions, possibly unbounded, to the
semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded
from below. Under the homogeneous Dirichlet boundary condition, we
prove new monotonicity results for $u$, when $f$ is a (locally or
globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As
an application of our new monotonicity theorems, we prove some
classification and/or non-existence results. Also, we answer a
question (raised by Berestycki, Caffarelli and Nirenberg) about
Serrin's overdetermined problems on epigraphs.