Remarks on the existence of branch bubbles on the blowup analysis of equation $- \Delta u = e^{2u}$ in dimension two
Remarks on the existence of branch bubbles on the blowup analysis of equation $- \Delta u = e^{2u}$ in dimension two
复制标题
二维方程$-Delta u = e^{2u}$的吹胀分析中分支气泡存在性的评述
DOI：
10.4310/cag.1999.v7.n2.a4
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发表时间：
1999
影响因子：
0.7
通讯作者：
Xiuxiong Chen
中科院分区：
文献类型：
--
作者：
Xiuxiong Chen
众所周知，对于在维数为3（参见[4]）或更高（参见[2]）的流形上具有常数标量曲率的固定共形类中的度量序列，爆破集必须是有限且简单的（每个冒泡点恰好携带一个球面）。人们普遍认为，维度2中的相应说法也是成立的。作者在研究一个不同但相关的问题[5]、[6]时，从李燕燕和Tai Shafrir [3]的联合论文中了解到这个问题。本文的目的是构造一个例子的一个序列的度量在一个单位圆盘具有恒定的曲率1和均匀有界的区域，开发分支气泡在中心的磁盘，从而提供一个反例类似的陈述在二维。感兴趣的读者可以参考[3]以了解这个问题的详细历史和进一步的参考资料。非常简单地说，Brezis和Merle [1]研究了满足以下方程的开单位圆盘Bi中的解序列{^n}：
It is well known that for a sequence of metrics in a fixed conformal class with constant scalar curvature on a manifold with dimension 3 (see [4] for further reference) or higher (see [2] for further reference), the blowingup set must be finite and simple (each bubbling point carry exactly one sphere). The corresponding statement in dimension 2 is widely expected to hold. The author learned this problem from a joint paper of YanYan Li and Tai Shafrir [3] when he studied a different but related problem [5], [6]. The purpose of this paper is to construct an example of a sequence of metrics in a unit disk with constant curvature 1 and uniformly bounded area which develops branch bubbles at the center of the disk, thereby providing an counter example to the analogous statements in dimension two. Interested readers are referred to [3] for detailed history of this problem and further references. Very briefly, Brezis and Merle [1] studied a sequence of solutions {^n} in an open unit disk Bi satisfying the equation: