On Conformally Invariant Equation ( -a )p U -k(x) U N-2, = 0 and Its Generalizations (*). Where 2 N 3-4n 2 + 16n-16 Cn = Dn = (n -2 )2 ' 8(n -1 )2 (n -2 )2

We consider the question of existence and non-existence of positive entire solutions for conformally invariant equations involving polyharmonic operator. We obtain existence of infinitely many positive solutions if the potential decays sufficiently fast at infinity and the nonexistence of positive solutions if the potential grows too fast at infinity. We also establish a Kazdan-Warner type condition for non-existence of solutions decaying at infinity. 1. I n t r o d u c t i o n . Let us start with the concept of conformally invariant operators. On a general Riemannian manifold M with metric g, a metrically defined operator A is said to be conformally invariant if metrics gw and g are pointwise conformally related, i.e., if gw = e2Wg, the pair of corresponding operators Aw and A are related by (1.1) Aw(qg) = e-bWA(eaWq)) for all c p ~ C ~ ( M ) . Conformal Laplacian 4(n 1 ) / (n 2 ) A k , where k is the scalar curvature of the metric g, is a well known second order conformally invariant operator. Associated with this well understood operator, there is a prescribed scalar curvature problem. Given a smooth positive function K defined on a Riemannian manifold (M, go) of dimension n I> 2, we ask whether there exists a metric g pointwise conformal to go such that K is the scalar curvature 4 of the new metric g. Let g = e2Ug o for n = 2 or g = u ,-2 go for n I-3, then the problem is reduced to find solutions of the following nonlinear elliptic equations: (1.2) A go u + Ke ~ = ko (*) Entrata in Redazione 1'11 novembre 1998. Ricevuta nuova versione 1'1 novembre 1999. Indirizzo degli AA.: GtsozrtEN Et3: Department of Mathematics, Wayne State University, Detroit, MI 48202. E-mail address: [email protected]; JtJNCrtENO WE1: Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong. E-mail address: [email protected]; XSNCWaNC Xu: Department of Mathematics, National University of Singapore, Singapore 119260, Republic of Singapore. E-mail address: [email protected] 310 GUOZHEN LU J U N C H E N G W E I XINGWANG XU: On conformally invariant, etc.