Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

Changshou Lin; Liping Wang; Juncheng Wei
October 2007
Calculus of Variations & Partial Differential Equations;Oct2007, Vol. 30 Issue 2, p153
Academic Journal
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 $${\partial\Omega;}$$ , O; 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 � K > 0 such that for � > � K , the above problem has K- bubble solution u � concentrating at the same point Q 0. More precisely, we show that u � has K local maximum points Q , ... , Q ??O 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 f ( Q) = Q T G Q with G = (? ij H( Q 0)).


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