Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q-curvature equation, we generalize the wellknown non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q= 1 solutions on Sn as well as on Rn with necessary growth rate assumption. © 2004 Elsevier Inc. All rights reserved.