In this short note, we study the prescribed Q-curvature equation with a singularity at the origin in \({\mathbb {R}}^4\), namely,

$$\begin{aligned} \Delta ^2u=(1-|x|^p)e^{4u}-c\delta _0\quad \text {in}\quad {\mathbb {R}}^4 \end{aligned}$$

under a finite volume condition, where \(p>0\) and \(c\in {\mathbb {R}}\). We prove the nonexistence of normal solutions to the above equation. This partly generalizes the nonexistence results of Hyder and Martinazzi (arXiv:2010.08987, 2020) where \(c=0\), and extends the conclusion of Hyder et al. (Int Math Res Not, 2019. https://doi.org/10.1093/imrn/rnz149) where \(p=0\).

中文翻译：

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奇异 Q 曲率方程的解不存在

在这个简短的笔记中，我们研究了原点在\({\mathbb {R}}^4\)处具有奇点的规定 Q 曲率方程，即，

$$\begin{aligned} \Delta ^2u=(1-|x|^p)e^{4u}-c\delta _0\quad \text {in}\quad {\mathbb {R}}^4 \结束{对齐}$$

在有限体积条件下，其中\(p>0\)和\(c\in {\mathbb {R}}\)。我们证明了上述方程的正态解不存在。这部分概括了 Hyder 和 Martinazzi (arXiv:2010.08987, 2020) 其中\(c=0\)的不存在结果，并扩展了 Hyder 等人的结论。(Int Math Res Not, 2019. https://doi.org/10.1093/imrn/rnz149) 其中\(p=0\)。