  In this paper, we are concerned with the following elliptic equation
\begin{equation*} \begin{cases} -\Delta u= Q(x)u^{2^*-1 }+\varepsilon u^{s},~
&{\text{in}~\Omega},\\[1mm] u>0,~ &{\text{in}~\Omega},\\[1mm] u=0,
&{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $N\geq 3$,
$s\in [1,2^*-1)$ with $2^*=\frac{2N}{N-2}$, $\varepsilon>0$, $\Omega$ is a
smooth bounded domain in $\mathbb{R}^N$.
  Under some conditions on $Q(x)$, Cao and Zhong in Nonlin. Anal. TMA (Vol 29,
1997, 461--483) proved that there exists a single-peak solution for small
$\varepsilon$ if $N\geq 4$ and $s\in (1,2^*-1)$. And they proposed in Remark
1.7 of their paper that
  \vskip 0.1cm\begin{center} \emph{``it is interesting to know the existence of
single-peak solutions for small $\varepsilon$ and $s=1$''.} \end{center}\vskip
0.1cm
  \noindent Also it was addressed in Remark 1.8 of their paper that
  \vskip 0.1cm \begin{center} \emph{``the question of solutions concentrated at
several points at the same time is still open''.} \end{center}\vskip 0.1cm
  \noindent
  Here we give some confirmative answers to the above two questions.
Furthermore, we prove the local uniqueness of the multi-peak solutions. And our
results show that the concentration of the solutions to above problem is
delicate whether $s=1$ or $s>1$.
