THE NUMBER OF POSITIVE SOLUTIONS TO THE BREZIS-NIRENBERG PROBLEM

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.

Automated summary

This paper focuses on the local uniqueness of multi-peak solutions in the well-known Brezis-Nirenberg problem and the exact number of positive solutions for small epsilon > 0. By using various local Pohozaev identities and blow-up analysis, the relationship between the profile of blow-up solutions and Green's function is investigated, leading to a type of local uniqueness results. Additionally, a description of the number of positive solutions for small positive epsilon is provided, which is dependent on Green's function.