Normal conformal metrics on [formula omitted] with Q-curvature having power-like growth

Abstract

Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg problem, we show that the prescribed Q-curvature equationΔ2u=(1−|x|p)e4u in R4,Λ:=∫R4(1−|x|p)e4udx<∞ has normal solutions (namely solutions which can be written in integral form, and hence satisfy Δu(x)=O(|x|−2) as |x|→∞) if and only if p∈(0,4) and(1+p4)8π2≤Λ<16π2.We also prove existence and non-existence results for the positive curvature case, namely for Δ2u=(1+|x|p)e4u in R4, and discuss some open questions.