On conformally invariant equation $$( - \Delta )^p u - K(x)u^{\tfrac{{N + 2p}}{{N - 2p}}} = 0$$ and its generalizationsand its generalizations

Abstract

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.