In this paper we are concerned with the well-known Brezis-Nirenberg problem
{
−
Δ
u
=
u
N
+
2
N
−
2
+
ε
u
,
a
m
p
;
in
Ω
,
u
>
0
,
a
m
p
;
in
Ω
,
u
=
0
,
a
m
p
;
on
∂
Ω
.
\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
ε
>
0
\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
ε
>
0
\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
Ω
\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
ε
\varepsilon
, which depends also on Green’s function.