Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

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Abstract

We consider the following critical elliptic Neumann problem ${- \Delta u+\mu u=u^{\frac{N+2}{N-2}}, u > 0 in \Omega; \frac{\partial u}{\partial n}=0}$ on ${\partial\Omega;}$ , Ω; being a smooth bounded domain in ${\mathbb{R}^{N}, N\geq 7, \mu > 0}$ 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 Kbubble 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 ${u_{\mu} (Q_j^\mu) \sim \mu^{\frac{N-2}{2}}, Q_j^\mu \to Q_0, j=1,\ldots , K,}$ and ${ \mu^{\frac{N-3}{N}} ((Q_1^{\mu})^{'}, \ldots , (Q_K^{\mu})^{'}) }$ approach an optimal configuration of the following functional

(*) Find out the optimal configuration that minimizes the following functional: ${R[Q_1^{'}, \ldots , Q_K^{'}]= c_1 \sum\limits_{j=1}^K \varphi (Q_j^{'}) + c_2 \sum\limits_{ i \not = j} \frac{1}{|Q_i^{'}-Q_j^{'}|^{N-2}}}$ where ${Q_i^\mu= ((Q_i^{\mu})^{'}, Q_{i, N}^\mu), c_1, c_2 > 0}$ are two generic constants and φ (Q) = Q T G Q with G = (∇ ij H(Q 0)).

Research supported in part by an Earmarked Grant from RGC of HK.