On the Brézis–Nirenberg Problem
On the Brézis–Nirenberg Problem
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DOI：
10.1007/s00205-009-0288-8
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发表时间：
2010-01
影响因子：
2.5
通讯作者：
Martin Schechter;W. Zou
中科院分区：
文献类型：
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作者：
Martin Schechter;W. Zou
本文研究了如下Brézis-Nirenberg问题（Comm Pure Appl Math 36：437-477，1983）：其中Ω是R的有界光滑区域 N（N ≠ 7）和2* 是临界Sobolev指数。我们证明了，对于每个固定的λ > 0，这个问题有无穷多个变号解。特别是，如果λ ≥ λ1，则Brézis-Nirenberg问题具有且仅具有除零之外的无限多个符号变化解。主要工具是节点解的莫尔斯指数的估计。
We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983):where Ω is a bounded smooth domain of R N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ1, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.