Normal conformal metrics on R4 with Q-curvature having power-like growth

Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg problem, we show that the prescribed Q-curvature equationDelta 2u = (1 - vertical bar x vertical bar(p))e(4u) in R-4, Lambda := integral(R4) (1 - vertical bar x vertical bar(p))e(4u) dx < infinityhas normal solutions (namely solutions which can be written in integral form, and hence satisfy Delta u(x) = O (vertical bar x vertical bar(-2)) as vertical bar x vertical bar ->infinity) if and only if p is an element of (0, 4) and(1 + p) 8 pi(2) <= < 16 pi(2).We also prove existence and non-existence results for the positive curvature case, namely for Delta(2)u = (1 + vertical bar x vertical bar(p))e(4u) in R-4, and discuss some open questions. (C) 2021 Elsevier Inc. All rights reserved.