Monotonicity for solutions to semilinear problems in epigraphs

Abstract

We consider positive solutions, possibly unbounded, to the semilinear equation $-\hat{I}”u=f(u)$ 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 $f(0)\ge 0$. 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 $\hat{I}\text{c◯}$. 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.