Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${- \Delta u+\mu u=u^{\frac{N+2}{N-2}}, u > 0 in \Omega; \frac{\partial u}{\partial n}=0}$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\partial\Omega;}$$\end{document} , Ω; being a smooth bounded domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}, N\geq 7, \mu > 0}$$\end{document} is a large number. We show that at a positive nondegenerate local minimum point Q0 of the mean curvature (we may assume that Q0 = 0 and the unit normal at Q0 is − eN) for any fixed integer K ≥ 2, there exists a μK > 0 such that for μ > μK, the above problem has K−bubble solution uμ concentrating at the same point Q0. More precisely, we show that uμ has K local maximum points Q1μ, ... , QKμ ∈∂Ω with the property that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_{\mu} (Q_j^\mu) \sim \mu^{\frac{N-2}{2}}, Q_j^\mu \to Q_0, j=1,\ldots , K,}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \mu^{\frac{N-3}{N}} ((Q_1^{\mu})^{'}, \ldots , (Q_K^{\mu})^{'}) }$$\end{document} approach an optimal configuration of the following functional