Title: Monotonicity and symmetry of nonnegative solutions to $ -Δu=f(u) $ in half-planes and strips,
Abstract: We consider nonnegative solutions to $-\Delta u=f(u)$ in half-planes and
strips, under zero Dirichlet boundary condition. Exploiting a
rotating$\&$sliding line technique, we prove symmetry and monotonicity
properties of the solutions, under very general assumptions on the nonlinearity
$f$. In fact we provide a unified approach that works in all the cases
$f(0)<0$, $f(0)= 0$ or $f(0)> 0$. Furthermore we make the effort to deal with
nonlinearities $f$ that may be not locally-Lipschitz continuous. We also
provide explicite examples showing the sharpness of our assumptions on the
nonlinear function $f$.