Nonlinear elliptic equations on the upper half space

Abstract

In this paper we shall classify all positive solutions of $\Delta u=a{u}^p$ on the upper half space $H={\mathbb{R}}_{+}^n$ with nonlinear boundary condition $\partial u/\partial t=-b{u}^q$ on $\partial H$ for both positive parameters $a,b>0$. We will prove that for $p\ge (n+2)/(n-2),1\le q<n/(n-2)$ (and $n\ge 3$) all positive solutions are functions of last variable; for $p=(n+2)/(n-2),q=n/(n-2)$ (and $n\ge 3$) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.