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 $${\Delta }^{2}u=(1-|x{|}^p)e^{4u}\text{ in }{\mathbb{R}}^4,\Lambda :={\int }_{{\mathbb{R}}^4}(1-|x{|}^p)e^{4u}dx<\infty$$ 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.

References in corpus (1)

• Classification of solutions to the higher order Liouville's equation on R^{2m}

Cited by in corpus (1)

• Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature