Multi-peak standing waves for nonlinear Schrodinger equations involving critical growth
Multi-peak standing waves for nonlinear Schrodinger equations involving critical growth
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涉及临界增长的非线性薛定谔方程的多峰驻波
DOI：
10.1002/mana.201500074
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发表时间：
2017
影响因子：
1
通讯作者：
Yang Minbo
中科院分区：
文献类型：
--
作者：
Zhang Jianjun;do O Joao Marcos;Yang Minbo
本文研究了具有临界增长的非线性薛定谔方程− ε2 v+ V（x）v= f（v），v> 0，v∈ H1（RN），（1.1）其中N≥ 3.从物理的角度来看，这些小ε的驻波被称为半经典态，它描述了从量子力学到经典力学的转变。在过去的几十年里，人们对问题（1.1）给予了相当大的关注。问题（1.1）的一类有趣的解是当ε→ 0时在RN中某个点周围形成尖峰形状的解族。相关结果参见[3]、[4]、[5]、[6]、[14]、[16]、[17]、[23]、[24]、[29]。在开创性的工作[18]中，对于N= 1和f（s）= s3，Floer和Weinstein构造了问题（1.1）的单峰解，其集中在V的任何给定的非退化临界点附近。|S| p− 2 s，p∈（2，2 <$），2 <$= 2N/（N− 2），N≥ 3，Oh [25]将[18]中的结果推广到高维情形.在[18]，[25]中，这些方法基于Lyapunov-Schmidt约化，其中基本上需要非简并条件。但非简并条件一般不容易检验。它是已知的，只有一个相当有限的一类非线性f。对于更一般的非线性问题，Rabinowitz [27]开创了用变分方法研究问题（1.1），并在全局位势条件下证明了问题（1.1）正解的存在性。|X| →∞
In this paper, we shall be concerned with the existence of positive solutions of the following nonlinear Schrödinger equation with critical growth− ε2 v+ V (x) v= f (v), v> 0, v∈ H1 ( RN),(1.1) where N≥ 3. From the physical point of view, these standing waves for small ε are referred as semi-classical states, which describe the transition from quantum mechanics to classical mechanics. In the last decades, there have been considerable attentions paid to problem (1.1). An interesting class of solutions of problem (1.1) are families of solutions which develop a spike shape around some certain point in RN as ε→ 0. For related results, we refer to [3],[4],[5],[6],[14],[16],[17],[23],[24],[29]. In the pioneering work [18], for N= 1 and f (s)= s3, Floer and Weinstein constructed a single-peak solution of problem (1.1), which concentrates around any given non-degenerate critical point of V. For f (s)=| s| p− 2 s, p∈(2, 2∗), 2∗= 2N/(N− 2), N≥ 3, Oh [25] extended the result in [18] to the higher dimensional case. In [18],[25], the methods were based on a Lyapunov–Schmidt reduction, in which a non-degeneracy condition is essentially required. But the non-degeneracy condition is not easy to check in general. It is known only for a rather restricted class of nonlinearities f. For more general nonlinearity, Rabinowitz [27] initiated the variational approach to study the problem (1.1) and proved the existence of positive solutions of (1.1) under the global potential condition lim inf| x|→∞