Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient

TL;DR

This paper investigates the monotonicity of positive solutions to a class of singular quasilinear elliptic problems in half-spaces, providing new insights even in classical cases and employing boundary behavior analysis and the moving plane method.

Contribution

It establishes the monotonicity of solutions in the $x_N$-direction for a broad class of singular quasilinear elliptic problems, including new results for $ heta=0$ and $p=2$ cases.

Findings

Solutions are monotone in the $x_N$-direction under specified conditions.

Behavior of solutions and derivatives near the boundary is characterized.

Most results are new even for classical cases with $ heta=0$ or $p=2$.

Abstract

We study positive solutions to the problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $\gamma>0$, $0<q\le p$, $\vartheta\ge0$ and $f:[0,+\infty)\to\mathbb{R}$ is a locally Lipschitz continuous function. We describe the behavior of solutions and their derivatives near the boundary. Then we exploit that information and the moving plane method to prove the monotonicity of solutions in the $x_N$-direction. This result holds for $W^{1,p}_{\rm loc}(\mathbb{R}^2_+)\cap L^\infty_{\rm loc}(\overline{\mathbb{R}^2_+})$ solutions in dimension two and for $W^{1,p}_{\rm loc}(\mathbb{R}^N_+)$ solutions which are bounded in strips in higher dimensions. Most of our results are new even in the case $\vartheta=0$ or $p=2$.