Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem Δ u+\mu u=u^{frac{N+2}{N-2}}, u > 0 in Ω u n}=0} on ℝ Ω , Ω; being a smooth bounded domain in ℝ^N, N ≥ 7, μ > 0 is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is - e _N ) for any fixed integer K 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K-bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁ ^μ , ... , Q _K ^μ Ω with the property that u_μ (Q_jμ ∼ μ_n, Q_j to Q ₀, j=1,...,K, and μ {N-3}{N}} ((Q_1 μ) {'},..., (Q_Kμ) approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: ℝ Q 1,..., QK= c_1 ∑ _j=1^K _j=1^K + c₂2 ∑ i = j {1}_j=1^K{N-2}} where Q_i ^μ= Q_i ^μ, Q_i ^μ, c_1, c_2 > 0} are two generic constants and φ (Q) = Q ^T G Q with G = (a_ij H(Q ₀)).