CLASSIFICATIONS OF NON-NEGATIVE SOLUTIONS TO SOME ELLIPTIC PROBLEMS

YUAN LOU MEIJUN ZHU

Abstract. The main purpose of this paper is to show that all non-negative solutions to Δu = 0 (or Δu = u^{p}) in the n-dimensional upper half space H = {(x′, t)|x′ ∈ ℝ^{n−1}, t > 0} with boundary condition ∂u/∂t = u^{q} on ∂H must be linear functions of t (or u ≡ 0) when n ≥ 2 and q > 1 (or n ≥ 2 and p, q > 1).