论文标题
$ \ mathbb {r}^4 $带有$ q $ curvature具有类似功率增长的普通共形指标
Normal conformal metrics on $\mathbb{R}^4$ with $Q$-curvature having power-like growth
论文作者
论文摘要
回答与Nirenberg问题中的爆炸行为有关的Struwe（越南J.Math。2020）的问题，我们表明处方$ Q $ -Curvature等式$$δ^2 u =（1- | x |^p） λ：= \ int _ {\ Mathbb {r}^4}（1- | x |^p） $ p \ in（0,4）$和$$ \ left（1+ \ frac {p} {4} \ right）8π^2 \leλ<16π^2。$$ 我们还证明了阳性曲率情况的存在和不存在结果，即$δ^2 u =（1+ | x |^p）e^{4u} $ in $ \ mathbb {r}^4 $，并讨论一些开放问题。
Answering a question by M. Struwe (Vietnam J. Math. 2020) related to the blow-up behaviour in the Nirenberg problem, we show that the prescribed $Q$-curvature equation $$Δ^2 u=(1-|x|^p)e^{4u}\text{ in }\mathbb{R}^4,\quad Λ:=\int_{\mathbb{R}^4}(1-|x|^p)e^{4u}dx<\infty$$ has normal solutions (namely solutions which can be written in integral form, and hence satisfy $Δu(x) =O(|x|^{-2})$ as $|x|\to \infty$) if and only if $p\in (0,4)$ and $$\left(1+\frac{p}{4}\right)8π^2\le Λ<16π^2.$$ We also prove existence and non-existence results for the positive curvature case, namely for $Δ^2 u=(1+|x|^p)e^{4u}$ in $\mathbb{R}^4$, and discuss some open questions.
