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Multi-bump analysis for Trudinger–Moser nonlinearities. I. Quantification and location of concentration points
Journal of the European Mathematical Society ( IF 2.6 ) Pub Date : 2020-09-16 , DOI: 10.4171/jems/1002
Olivier Druet 1 , Pierre-Damien Thizy 1
Affiliation  

In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving existence results of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what had been done for the Liouville equation in the beautiful work of Chen-Lin [8].

中文翻译:

Trudinger-Moser 非线性的多凸点分析。一、集中点的量化和定位

在本文中,我们仔细研究了一些椭圆偏微分方程的解序列在二维中包含非线性与特鲁丁格-莫泽增长的爆炸行为。第一作者在[15]中已经获得了量化结果,但仍有许多问题悬而未决。在随后的论文中也明确提出了类似的问题,参见 Del Pino-Musso-Ruf [12]、Malchiodi-Martinazzi [30] 或 Martinazzi [34]。我们回答了所有这些问题,特别证明了由于该方程中气泡之间的强相互作用,吹胀现象非常具有限制性。这项工作将有一个续集,本着陈林 [8] 的美丽工作中对刘维尔方程所做的精神,通过度数论论证给出了所有能级相关泛函临界点的存在结果。
更新日期:2020-09-16
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