Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent
 Changshou Lin,
 Liping Wang,
 Juncheng Wei
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Abstract
We consider the following critical elliptic Neumann problem ${ \Delta u+\mu u=u^{\frac{N+2}{N2}}, u > 0 in \Omega; \frac{\partial u}{\partial n}=0}$ 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 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 ${u_{\mu} (Q_j^\mu) \sim \mu^{\frac{N2}{2}}, Q_j^\mu \to Q_0, j=1,\ldots , K,}$ and ${ \mu^{\frac{N3}{N}} ((Q_1^{\mu})^{'}, \ldots , (Q_K^{\mu})^{'}) }$ approach an optimal configuration of the following functional
(*) Find out the optimal configuration that minimizes the following functional: ${R[Q_1^{'}, \ldots , Q_K^{'}]= c_1 \sum\limits_{j=1}^K \varphi (Q_j^{'}) + c_2 \sum\limits_{ i \not = j} \frac{1}{Q_i^{'}Q_j^{'}^{N2}}}$ 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})).
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 Title
 Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent
 Journal

Calculus of Variations and Partial Differential Equations
Volume 30, Issue 2 , pp 153182
 Cover Date
 20071001
 DOI
 10.1007/s0052600600825
 Print ISSN
 09442669
 Online ISSN
 14320835
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Primary: 35B40
 35J20
 Secondary: 35J55
 92C15
 92C40
 Authors

 Changshou Lin ^{(1)}
 Liping Wang ^{(2)}
 Juncheng Wei ^{(2)}
 Author Affiliations

 1. Department of Mathematics, National Chung Cheng University, Minghsiung, Chia Yi, Taiwan
 2. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong