Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces

TL;DR

This paper proves that positive solutions to a class of quasilinear elliptic equations with a first-order term in half-spaces are monotone increasing orthogonal to the boundary, using a new comparison principle and deriving Liouville theorems.

Contribution

Introduces a new comparison principle for unbounded domains and establishes monotonicity and Liouville theorems for quasilinear problems in half-spaces.

Findings

Positive solutions are monotone increasing orthogonal to the boundary.

Established new Liouville type theorems for these equations.

Developed a novel comparison principle for unbounded domains.

Abstract

We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.