Concentrated solutions for a critical elliptic equation

In this paper, we are concerned with the following elliptic equation <tex-math id="FE1"> {document}$ {equation*} {cases} -Δ u = Q(x)u2^*-1+ε uˢ, \; &{\;{in}\;\; Ω},\\ \ u>0, \; &{\;{in}\;\; Ω}, \\ \ u = 0, &{\;{on}\;\; ∂ Ω}, {cases} {equation*} document </tex-math> where <tex-math id="M1">document N≥ 3 document</tex-math>, <tex-math id="M2">document s∈ [1, 2^*-1) document</tex-math> with <tex-math id="M3">document 2^* = 2N/N-2 document</tex-math>, <tex-math id="M4">document ε>0 document</tex-math>, <tex-math id="M5">document Ω document</tex-math> is a smooth bounded domain in <tex-math id="M6">document R^N document</tex-math>. Under some conditions on <tex-math id="M7">document Q(x) document</tex-math>, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997,461–483) proved that there exists a single-peak solution for small <tex-math id="M8">document ε document</tex-math> if <tex-math id="M9">document N≥ 4 document</tex-math> and <tex-math id="M10">document s∈ (1, 2^*-1) document</tex-math>. And they proposed in Remark 1.7 of their paper that <tex-math id="FE2"> document {array}{c} ''it\; is\; interesting\; to\; know\; the \;existence\; of\; single-peak \;solutions \; for \;small \;ε\;\\ and \;s = 1''.{array} document </tex-math> Also it was addressed in Remark 1.8 of their paper that <tex-math id="FE3"> document {array}{c} ''the\; question\; of \;solutions \;concentrated \;at \;several\; points \;at \;the \;same \;time \;is \\ still \;open''. {array}${document} </tex-math> Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether <tex-math id="M11">{document}$ s = 1 ${document}</tex-math> or <tex-math id="M12">{document}$ s>1 ${document}</tex-math>.