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Concentration phenomena to a higher order Liouville equation
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-07-12 , DOI: 10.1007/s00526-020-01788-4
Ali Hyder

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 \({\tilde{\Omega }}\Subset (\Omega \setminus S_\varphi )\).



中文翻译:

高阶Liouville方程的集中现象

我们研究给定Q曲率问题的一系列解\((u_k)\)的爆炸和量化现象

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

\(Q_k \)的自然假设下。众所周知,直到一个子序列,\((u_k)\)都以合适的范数为界,或者存在\(\ beta _k \ rightarrow \ infty \)使得\(u_k = \ beta _k (\ varphi + O(1))\)\(\欧米茄\ setminus(S_1 \杯S_ \ varphi)\)对于一些非平凡非正ñ K谐波函数\(\ varphi \)和对于有限集\(S_1 \) ,其中\(S_ \ varphi \)是零组\(\ varphi \) 。我们证明了全曲率的量化\(\ INT _ {{\波浪{\欧米茄}}} Q_ke ^ {} 2nu_k DX \)的区域\({\波浪号{\ Omega}} \子集(\ Omega \ setminus S_ \ varphi)\)

更新日期:2020-07-13
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