The number of positive solutions to the Brezis-Nirenberg problem

Cao, Daomin; Luo, Peng; Peng, Shuangjie

doi:10.1090/tran/8287

The number of positive solutions to the Brezis-Nirenberg problem
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by Daomin Cao, Peng Luo and Shuangjie Peng PDF

Trans. Amer. Math. Soc. 374 (2021), 1947-1985 Request permission

Abstract:

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin {cases} -\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, &{\text {in}~\Omega },\\ u>0, &{\text {in}~\Omega },\\ u=0, &{\text {on}~\partial \Omega }. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon >0$ 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 $\varepsilon >0$.

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 $\Omega$ 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 $\varepsilon$, which depends also on Green’s function.

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