Annali di Matematica Pura ed Applicata ( IF 1 ) Pub Date : 2021-06-12 , DOI: 10.1007/s10231-021-01130-5 Shaowei Chen , Zhi-Qiang Wang
In Chen and Wang (Calc Var Part Differ Equ 56:1–26, 2017), we show that, if \(\epsilon >0\) is small enough, then there exists a sequence of semiclassical states of higher topological type localized at a local minimum set of the potential V for the semiclassical nonlinear Schrödinger equation
$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1\left( {\mathbb{R}}^N\right) . \end{aligned}$$In this paper, we consider a situation where V has multiple isolated local minimum sets. We show that as \(\epsilon \rightarrow 0,\) there exist multi-bump solutions of this equation being concentrated at those given local minimum sets while at the same time each bump behaves as a higher-topological-type solution in one local minimum set as aforementioned. Thus, the multi-bump solutions given here are constructed by gluing a sum of higher-topological-type solutions localized in separated local minimal sets of the potential.
中文翻译：
非线性薛定谔方程的高拓扑类型半经典状态的粘合
在 Chen 和 Wang (Calc Var Part Differ Equ 56:1–26, 2017) 中，我们表明，如果\(\epsilon >0\)足够小，那么存在定位于半经典非线性薛定谔方程的势V的局部最小值集
$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1\left( {\mathbb{R}} ^N\对）。\end{对齐}$$在本文中，我们考虑V有多个孤立的局部最小值集的情况。我们证明\(\epsilon \rightarrow 0,\)存在该方程的多凸点解集中在那些给定的局部最小值集上，同时每个凸点在一个局部中表现为更高拓扑类型的解如上所述的最小设置。因此，这里给出的多凸点解决方案是通过粘合位于分离的局部最小势集中的更高拓扑类型的解决方案的总和来构建的。
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