Normal conformal metrics on $\mathbb{R}^4$ with $Q$-curvature having power-like growth

Abstract

Answering a question by M. Struwe (Vietnam J. Math. 2020) related to the blow-up behaviour in the Nirenberg problem, we show that the prescribed $Q$-curvature equation $$\Delta^2 u=(1-|x|^p)e^{4u}\text{ in }\mathbb{R}^4,\quad \Lambda:=\int_{\mathbb{R}^4}(1-|x|^p)e^{4u}dx<\infty$$ has normal solutions (namely solutions which can be written in integral form, and hence satisfy $\Delta u(x) =O(|x|^{-2})$ as $|x|\to \infty$) if and only if $p\in (0,4)$ and $$\left(1+\frac{p}{4}\right)8\pi^2\le \Lambda <16\pi^2.$$ We also prove existence and non-existence results for the positive curvature case, namely for $\Delta^2 u=(1+|x|^p)e^{4u}$ in $\mathbb{R}^4$, and discuss some open questions.