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

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation $-\mathrm{\Delta }u=f(u)$ in the upper half-space ${\mathbb{R}}_{+}^N$. Some Liouville-type theorems are also proven in the case of differential inequalities in ${\mathbb{R}}_{+}^N$, 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 $f$ that behaves at least linearly at infinity.