Quasilinear elliptic problems with singular nonlinearities in half-spaces

TL;DR

This paper investigates positive solutions to quasilinear elliptic equations with singular nonlinearities in half-spaces, focusing on their symmetry, monotonicity, and classification, especially for singular power-type nonlinearities.

Contribution

It establishes symmetry and monotonicity results for solutions and classifies solutions for specific singular nonlinearities, advancing understanding of such problems in half-spaces.

Findings

Solutions exhibit monotonicity and one-dimensional symmetry.

Classification results for singular nonlinearities like 1/u^γ.

Extension of known results to p-Laplacian with singular terms.

Abstract

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally Lipschitz continuous function with a possible singularity at zero. Classification results for the case $f(u)=\frac{1}{u^\gamma}$ with $\gamma>0$ are also provided.