  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 $p \ge {(n+2)}/{(n-2)}, 1\leq
q<{n}/{(n-2)}$ (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.
