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Nonexistence of solutions to singular Q-curvature equations

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Abstract

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 \(p=0\).

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Acknowledgements

The author is very grateful to the referee for his careful reading and valuable suggestions on the first version of this paper.

Funding

This work was supported by the Outstanding Innovative Talents Cultivation Funded Programs 2020 of Renmin University of China.

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  1. Department of Mathematics, Renmin University of China, Beijing, 100872, China

    Yamin Wang

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  1. Yamin Wang
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Correspondence to Yamin Wang.

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Wang, Y. Nonexistence of solutions to singular Q-curvature equations. Arch. Math. 117, 455–467 (2021). https://doi.org/10.1007/s00013-021-01644-7

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