Classification of Positive Solutions to a Divergent Equation on the Upper Half Space

In this paper we classify the positive solutions of the divergent equation with Neumann boundary on the upper half space {-div(t(alpha) del u) = t(beta) f(u), (y, t) is an element of R-+(n+1), lim(t -> 0+) t(alpha) partial derivative u/partial derivative t = 0 by the method of moving spheres and Kelvin transformations, where n >= 1, alpha > 0, beta > -1, n-1/n+1 beta <= alpha > beta + 2 and f: (0, infinity) -> (0, infinity) is non-negative continuous function satisfying some conditions. This equation arises from a weighed Sobolev inequality involving divergent operator div(t(alpha) del u) on the upper half space.