One-dimensional symmetry results for semilinear equations and inequalities on half-spaces

We consider non-negative solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ in the upper half-space $R^N_+$ and we prove new one-dimensional symmetry results. Some Liouville-type theorems are also proven in the case of differential inequalities in $R^N_+$, even without imposing any boundary condition. Although subject to dimensional restrictions, our results apply to a broad family of functions $f$. In particular, they apply to all non-negative function $f$ that behaves at least linearly at infinity.