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.
Categories: math.AP