Quasilinear elliptic problems with singular nonlinearities in half-spaces

Abstract

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\hat{I}{”}_{p}u=f(u)$ 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 $f(u)=\frac{1}{u^3}$ with ${\hat{I}}^3>0$ are also provided.

34 pages