Monotonicity and Liouville-type theorems for semilinear elliptic problems in the half space

We consider classical solutions to $-\mathrm{\Delta }u=f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.