Monotonicity for solutions to semilinear problems in epigraphs

TL;DR

This paper establishes new monotonicity, classification, and non-existence results for positive solutions of semilinear equations in epigraphs, using comparison principles and a modified moving plane method.

Contribution

It introduces novel comparison principles and a modified moving plane technique tailored for unbounded epigraph domains, advancing the analysis of semilinear PDE solutions.

Findings

Proved new monotonicity results for solutions in epigraphs.

Established comparison principles for unbounded open sets.

Derived uniqueness and symmetry results for solutions in general unbounded domains.

Abstract

We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when $f$ is a (locally or globally) Lipschitz-continuous function satisfying $ f(0) \geq 0$. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of $\mathbb{R}^N$, and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph $\Omega$. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains.