Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

Original Article

DOI: 10.1007/s00526-006-0082-5

Cite this article as:
Lin, C., Wang, L. & Wei, J. Calc. Var. (2007) 30: 153. doi:10.1007/s00526-006-0082-5

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 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 Kbubble 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 \({u_{\mu} (Q_j^\mu) \sim \mu^{\frac{N-2}{2}}, Q_j^\mu \to Q_0, j=1,\ldots , K,}\) 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) = QTGQ with G = (∇ijH(Q0)).

Mathematics Subject Classification (2000)

Primary: 35B4035J20Secondary: 35J5592C1592C40

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsNational Chung Cheng UniversityChia YiTaiwan
  2. 2.Department of MathematicsChinese University of Hong KongShatinHong Kong