Nonlinear elliptic equations on the upper half space

In this paper, we shall classify all positive solutions of [Formula: see text] on the upper half space [Formula: see text] with nonlinear boundary condition [Formula: see text] on [Formula: see text] for parameters [Formula: see text] and [Formula: see text]. We will prove that for [Formula: see text] or [Formula: see text], [Formula: see text] (and [Formula: see text]) all positive solutions are functions of last variable; for [Formula: see text] (and [Formula: see text]) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.