In this paper we consider the fractional nonlinear Schrödinger equation ε(−∆)v + V (x)v = f(v), x ∈ R where s ∈ (0, 1), N ≥ 2, V ∈ C(RN ,R) is a positive potential and f is a nonlinearity satisfying Berestycki-Lions type conditions. For ε > 0 small, we prove the existence of at least cupl(K) + 1 positive solutions, where K is a set of local minima in a bounded potential well and cupl(K) denotes the cup-length of K. By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of K for ε small.

中文翻译：

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关于分数阶 NLS 方程和势井拓扑的影响

在本文中，我们考虑分数阶非线性薛定谔方程 ε(−∆)v + V (x)v = f(v), x ∈ R 其中 s ∈ (0, 1), N ≥ 2, V ∈ C(RN , R) 是正电位，f 是满足 Berestycki-Lions 类型条件的非线性。对于 ε > 0 小，我们证明至少存在 cupl(K) + 1 个正解，其中 K 是有界势阱中的一组局部最小值，cupl(K) 表示 K 的杯长。在变分方法中，我们分析了预期解邻域中不定泛函的两个级别之间的拓扑差异。由于非局域性直接来自空间的分解，我们通过合适的半范数引入了一个新的分数质心。一些其他微妙方面的出现与非本地运算符的存在严格相关。