The Asymptotic Behavior and Symmetry of Positive Solutions to p-Laplacian Equations in a Half-Space

We study a nonlinear equation in the half-space with a Hardy potential, specifically, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - {\Delta _p}u = \lambda {{{u^{p - 1}}} \over {x_1^p}} - x_1^\theta f\left(u \right)\,\,\,\,{\rm{in}}\,\,T,$$\end{document} where Δp stands for the p-Laplacian operator defined by Δpu = div(∣Δu∣p−2Δu), p > 1, θ > −p, and T is a half-space {x1 > 0}. When λ > Θ (where Θ is the Hardy constant), we show that under suitable conditions on f and θ, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as x1 → 0+, and the symmetric property of the positive solution are obtained.