The mass concentration phenomenon for L-2-critical constrained problems related to Kirchhoff equations
The mass concentration phenomenon for L-2-critical constrained problems related to Kirchhoff equations
复制标题
与基尔霍夫方程相关的L-2临界约束问题的质量集中现象
DOI：
10.1007/s00033-016-0624-4
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发表时间：
2016
影响因子：
2
通讯作者：
Ye Hongyu
中科院分区：
文献类型：
--
作者：
Ye Hongyu
本文研究了泛函I（u）= a2 <$R^ N的极大极小化临界点的集中性|u| ^ 2+ B 4\left（\int R^ N| u| ^ 2\right）^ 2-N 2N+ 8 <$R^ N| u| ^ 2N+ 8 N I（u）= a 2 RN| u| 2+ B 4| u| 2 2-N 2 N+ 8 RN| u| S_c={u ∈ H^ 1（R^ N）}上的2N + 8 N约束|~|u|_2= c，c> 0\} Sc = u∈ H1（RN）||u|当c →（c^*）^+ c→（c）+时，2= c，c> 0，其中c^*=\left（2^-1 B| Q|_2^ 8 N\right）^ N 8-2N c = 2-1 B| Q| 2 8 NN 8-2 N，N= 1，2，3，N= 1，2，3，且Q Q是平移的，-2 Δ Q+\left（4 N-1\right）Q=| Q| ^ 8 N Q-2 Δ Q+ 4 N-1 Q=| Q| 8 NQ在R^ N RN中。由于这类约束问题是L^2L2-临界的，因此似乎不可能从自然约束V_c=\left {u ∈ S_c| ~ a R^ N| u| ^ 2+ B\left（\int R^ N| u| ^2\right）^ 2= 2N N+ 4 <$R^ N| u| ^ 2N+ 8 N\right\} V c= u∈ S c|一名女律师|u| 2+ B级|u| 2 2= 2 NN+ 4 ðRN| u| 2 N+ 8 N。本文证明了对于某个子流形M_c <$V_c M_c <$V_c，山路能级γ（c）=\infu ∈ M_c I（u）γ（c）= infu ∈ M_c I（u），并证明了γ（c）γ（c）在（c^*，+ ∞）（c ^，+∞）上的严格单调性.我们得到了临界点u_c uc的性质类似于u_c（x）（a^ 2 2b（c^*）^ 2（c c^*）^ 8-2N N-1^2\right）^ N 8 Q\left（\left（a B（c^*）^ 2（c c^*）^ 8-2N N-1\right）^ 1 2（x-y_c）\right）uc（x）a 2 2 2 B（c）2（cc）8-2 NN-1 2 N 8 Q ab（c）2（cc）8 -2 NN-1 1 1 2（x-yc）对于某些y_c ∈ R^当c c从上面接近c^* c时，N yc∈ RN。
In this paper, we study the concentration behavior of critical points with a minimax characterization to the following functional I (u)= a 2 ∫ R^ N| ∇ u|^ 2+ b 4\left (\int R^ N| ∇ u|^ 2\right)^ 2-N 2N+ 8 ∫ R^ N| u|^ 2N+ 8 N I (u)= a 2∫ RN|∇ u| 2+ b 4∫ RN|∇ u| 2 2-N 2 N+ 8∫ RN| u| 2 N+ 8 N constrain on S_c={u ∈ H^ 1 (R^ N)|~| u| _2= c, c> 0\} S c= u∈ H 1 (RN)|| u| 2= c, c> 0 when c → (c^*)^+ c→(c∗)+, where c^*=\left (2^-1 b| Q| _2^ 8 N\right)^ N 8-2N c∗= 2-1 b| Q| 2 8 NN 8-2 N, N= 1, 2, 3, N= 1, 2, 3, and Q Q is up to translations, the unique positive solution of-2 Δ Q+\left (4 N-1\right) Q=| Q|^ 8 N Q-2 Δ Q+ 4 N-1 Q=| Q| 8 NQ in R^ N RN. As such constraint problem is L^ 2 L 2-critical, it seems impossible to benefit from natural constraints V_c=\left {u ∈ S_c|~ a ∫ R^ N| ∇ u|^ 2+ b\left (\int R^ N| ∇ u|^ 2\right)^ 2= 2N N+ 4 ∫ R^ N| u|^ 2N+ 8 N\right\} V c= u∈ S c| a∫ RN|∇ u| 2+ b∫ RN|∇ u| 2 2= 2 NN+ 4∫ RN| u| 2 N+ 8 N. We show that the mountain pass energy level γ (c)=\inf u ∈ M_c I (u) γ (c)= inf u∈ M c I (u) for some submanifold M_c ⊂ V_c M c⊂ V c and then prove the strict monotonicity of γ (c) γ (c) on (c^*,+ ∞)(c∗,+∞). We obtained that the critical point u_c uc behaves like u_c (x) ≈\left (a^ 2 2b (c^*)^ 2 (c c^*)^ 8-2N N-1^ 2\right)^ N 8 Q\left (\left (a b (c^*)^ 2 (c c^*)^ 8-2N N-1\right)^ 1 2 (x-y_c)\right) uc (x)≈ a 2 2 b (c∗) 2 (cc∗) 8-2 NN-1 2 N 8 Q ab (c∗) 2 (cc∗) 8-2 NN-1 1 2 (x-yc) for some y_c ∈ R^ N yc∈ RN as c c approaches c^* c∗ from above.