Positive solutions for a critical elliptic equation

Abstract: 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 $s>1$.