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Concentration phenomena to a higher order Liouville equation

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Abstract

We study blow-up and quantization phenomena for a sequence of solutions \((u_k)\) to the prescribed Q-curvature problem

$$\begin{aligned} (-\Delta )^nu_k= Q_ke^{2nu_k}\quad \text {in }\Omega \subset {\mathbb {R}}^{2n},\quad \int _{\Omega }e^{2nu_k}dx\le C, \end{aligned}$$

under natural assumptions on \(Q_k\). It is well-known that, up to a subsequence, either \((u_k)\) is bounded in a suitable norm, or there exists \(\beta _k\rightarrow \infty \) such that \( u_k=\beta _k(\varphi +o(1))\) in \(\Omega \setminus (S_1\cup S_\varphi )\) for some non-trivial non-positive n-harmonic function \(\varphi \) and for a finite set \(S_1\), where \(S_\varphi \) is the zero set of \(\varphi \). We prove quantization of the total curvature \(\int _{{\tilde{\Omega }}}Q_ke^{2nu_k}dx\) on the region \({\tilde{\Omega }}\Subset (\Omega \setminus S_\varphi )\).

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Acknowledgements

The author is greatly thankful to Luca Martinazzi and Pierre-Damien Thizy for various stimulating discussions.

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  1. Johns Hopkins University, Baltimore, Maryland, 21218, USA

    Ali Hyder

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  1. Ali Hyder
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Correspondence to Ali Hyder.

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Communicated by A. Chang.

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The author is supported by the Swiss National Science Foundation projects No. P2BSP2-172064 and P400P2-183866.

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Hyder, A. Concentration phenomena to a higher order Liouville equation. Calc. Var. 59, 126 (2020). https://doi.org/10.1007/s00526-020-01788-4

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