Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

TL;DR

This paper establishes a weak comparison principle for certain quasilinear elliptic equations in unbounded domains, leading to monotonicity results and Liouville-type theorems for solutions in half-spaces.

Contribution

It introduces a novel weak comparison principle in narrow unbounded domains for p-Laplacian equations, enabling new monotonicity and Liouville theorems.

Findings

Proved a weak comparison principle for $- abla_p u=f(u)$ in unbounded domains.

Established monotonicity of positive solutions in half-spaces.

Derived Liouville-type theorems for solutions under specified conditions.

Abstract

We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta_p u=f(u)$ in the case $2<p< 3$ and $f(\cdot)$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot)$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.