THE NUMBER OF POSITIVE SOLUTIONS TO THE BREZIS-NIRENBERG PROBLEM

作者：Cao, Daomin^*; Luo, Peng; Peng, Shuangjie

来源：Transactions of the American Mathematical Society, 2021, 374(3): 1947-1985.

DOI：10.1090/tran/8287

• Critical Sobolev exponent
• local Pohozaev identity
• existence of solutions
• exact number of solutions
• Green's function

In this paper we are concerned with the well-known Brezis-Nirenberg problem @@@ {-Delta u = u(N+2/N-2) + epsilon u, in Omega, @@@ u > 0, in Omega, @@@ u = 0, on partial derivative Omega, @@@ The existence of multi-peak solutions to the above problem for small epsilon > 0 was obtained by Musso and 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 epsilon > 0. @@@ 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 Q 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 epsilon, which depends also on Green's function.

• 单位
  中国科学院; 中国科学院研究生院