  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 by Musso and Pistoia.
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 the
Green's function of the domain $\Omega$ and then obtain a type of local
uniqueness results of blow-up solutions. At last we give a description of the
number of positive solutions for small positive $\varepsilon$, which depends
also on the Green's function.
