论文标题
球在第一个关键指数上的径向爆炸的分类
Classification of radial blow-up at the first critical exponent for the Lin-Ni-Takagi problem in the ball
论文作者
论文摘要
我们调查了球$ b_r \ subset \ mathbb {r}^n $ for $ n \ ge 3 $ 3 $：\ begin {equation*} \ left \ left \ okent {aligned} - \ triangle u_p + _p + _p＆w _ p＆w y_p， } b_r，\\ \ \partial_νu_p＆= 0＆\ textrm {on} \ partial b_r，\ end {aligned} \ right。 \ end {equation*}当$ p $接近第一个关键sobolev endent $ 2^* = \ frac {2n} {n-2} $时。我们获得了有限的能量径向平滑爆炸解决方案的完整分类。我们将防止爆炸的条件描述为$ p \至2^*$，我们提供必要的条件以进行爆炸，并通过构建炸毁序列的示例来确定它们的清晰度。我们的方法允许$ p $的渐近超临界值。我们特别表明，如果$ p \ geq 2^\ ast $，则有限能源径向解决方案在$ c^2（\ bar {b_r}）$中进行了预发，规定$ n \ geq 7 $。如果$ p = 2^\ ast $，也可以在较小的维度中给出足够的条件。最后，我们将我们的结果与非线性肛门的Bonheure，Grumiau和Troestler的分叉分析进行了比较。 147（2016）。
We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R \subset \mathbb{R}^N$ for $N \ge 3$: \begin{equation*} \left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R, \\ \partial_νu_p & = 0 & \textrm{ on } \partial B_R, \end{aligned} \right. \end{equation*} when $p $ is close to the first critical Sobolev exponent $2^* = \frac{2N}{N-2}$. We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as $p \to 2^*$, we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of $p$. We show in particular that, if $p \geq 2^\ast$, finite-energy radial solutions are precompact in $C^2(\bar{B_R})$ provided that $N\geq 7$. Sufficient conditions are also given in smaller dimensions if $p=2^\ast$. Finally we compare and interpret our results to the bifurcation analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).
