Concentrated solutions for a critical elliptic equation

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}, \; &{\;{\rm{in}}\;\; \Omega},\\ \ u>0, \; &{\;{\rm{in}}\;\; \Omega}, \\ \ u = 0, &{\;{\rm{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

$ \begin{array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;\varepsilon\;\\ and \;s = 1''.\end{array} $

Also it was addressed in Remark 1.8 of their paper that

$ \begin{array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. \end{array}$

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 $.

Correction: “Technology Foundation of Guizhou Province” is corrected to “The Science and Technology Foundation of Guizhou Province" under Fund Project.