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The number of positive solutions to the Brezis-Nirenberg problem
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2021-01-12 , DOI: 10.1090/tran/8287 Daomin Cao , Peng Luo , Shuangjie Peng
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2021-01-12 , DOI: 10.1090/tran/8287 Daomin Cao , Peng Luo , Shuangjie Peng
Abstract:In this paper we are concerned with the well-known Brezis-Nirenberg problem
The existence of multi-peak solutions to the above problem for small
was obtained (see Monica Musso and Angela Pistoia [Indiana Univ. Math. J. 51 (2002), pp. 541-579]). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small 
. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green's function of the domain 
and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive 
, which depends also on Green's function.
中文翻译：
Brezis-Nirenberg问题的正解数
摘要：本文关注的是著名的Brezis-Nirenberg问题
已经获得了针对上述小问题的多峰解决方案的存在（请参阅Monica Musso和Angela Pistoia [Indiana Univ。Math。J. 51（2002），第541-579页））。但是，上述问题的正解的唯一性或确切数目仍然未知。在这里，我们着重于多峰解的局部唯一性以及针对小问题的上述问题的正解的确切数目。通过使用各种局部Pohozaev身份和爆炸分析，我们首先检测爆炸解决方案的轮廓与域的格林函数之间的关系，然后获得一类局部唯一性的爆炸解决方案结果。最后，我们对小正数的正解数进行描述，这也取决于格林函数。
更新日期：2021-02-18
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The existence of multi-peak solutions to the above problem for small
was obtained (see Monica Musso and Angela Pistoia [Indiana Univ. Math. J. 51 (2002), pp. 541-579]). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small . By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green's function of the domain and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive , which depends also on Green's function. - [1]
中文翻译：
Brezis-Nirenberg问题的正解数
摘要：本文关注的是著名的Brezis-Nirenberg问题
|
已经获得了针对上述小问题的多峰解决方案的存在（请参阅Monica Musso和Angela Pistoia [Indiana Univ。Math。J. 51（2002），第541-579页））。但是，上述问题的正解的唯一性或确切数目仍然未知。在这里，我们着重于多峰解的局部唯一性以及针对小问题的上述问题的正解的确切数目。通过使用各种局部Pohozaev身份和爆炸分析，我们首先检测爆炸解决方案的轮廓与域的格林函数之间的关系，然后获得一类局部唯一性的爆炸解决方案结果。最后，我们对小正数的正解数进行描述，这也取决于格林函数。
- [1]
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