In this paper we study the semiclassical nonlinear Schrödinger equation

$$\begin{aligned} -\varepsilon ^2\Delta v+V(x)v=f(v),\quad x\in {\mathbb {R}}^N. \end{aligned}$$

where \(V\in C^1({\mathbb {R}}^N)\) is bounded and bounded away from zero and f is odd in v, subcritical and satisfies a monotonicity condition. We combine the del Pino and Felmer’s penalization approach and the Nehari manifold approach developed in Szulkin and Weth (J Funct Anal 257, 3802–3822, 2009; Handbook of nonconvex analysis and applications, International Press, Boston, pp 597–632, 2010) to obtain multiple localized solutions concentrating at the set of critical points of V(x).

中文翻译：

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半经典非线性Schrödinger方程的高拓扑类型的多重解

在本文中，我们研究了半经典非线性Schrödinger方程

$$ \ begin {aligned}-\ varepsilon ^ 2 \ Delta v + V（x）v = f（v），\ quad x \ in {\ mathbb {R}} ^ N。\ end {aligned} $$

其中\（V \ in C ^ 1（{\ mathbb {R}} ^ N）\）是有界且有界的，并且远离零，并且f在v中是奇数，亚临界且满足单调性条件。我们将del Pino和Felmer的罚分方法与在Szulkin和Weth中开发的Nehari流形方法相结合（J Funct Anal 257，3802–3822，2009；非凸分析和应用手册，国际出版社，波士顿，pp 597–632，2010）以获得集中在V（x）临界点集合上的多个局部解。