Nonlinear elliptic equations on the upper half space

In this paper we shall classify all positive solutions of $ Delta u =a u^p$ on the upper half space $ H =Bbb{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 $p ge {(n+2)}/{(n-2)}, 1leq q<{n}/{(n-2)}$ (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.