Positive solutions for a critical elliptic equation

Abstract

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -Δu= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~Ω},\\[1mm] u>0,~ &{\text{in}~Ω},\\[1mm] u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\ge 3$, $s\in [1,2^{\ast }-1)$ with $2^{\ast }=\frac{2N}{N-2}$, $\epsilon >0$, $\Omega$ 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 $\epsilon$ if $N\ge 4$ and $s\in (1,2^{\ast }-1)$. 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 $\epsilon$ 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 $s>1$.