Monotonicity and Liouville-type theorems for semilinear elliptic problems in the half space
Berardino Sciunzi, Domenico Vuono
TL;DR
The paper proves that positive solutions to $- abla^2 u = f(u)$ in a half-space with zero Dirichlet data are strictly monotone in the normal direction, assuming the solution is directionally bounded on finite strips (up to rotations). The authors adapt and refine the sliding-rotating moving-plane method, leveraging a boundary Harnack inequality and narrow-domain comparisons to propagate near-boundary monotonicity to the entire half-space. A central corollary is a Liouville-type nonexistence result for the Lane-Emden equation, showing no nontrivial positive solutions under the stated boundedness condition. The work strengthens previous monotonicity and Liouville-type results by relaxing boundedness assumptions and providing a higher-dimensional, rotation-compatible framework. These results enhance understanding of qualitative properties and nonexistence phenomena for semilinear elliptic equations in unbounded domains.
Abstract
We consider classical solutions to $-Δu = f(u)$ 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.