On the Fractional NLS Equation and the Effects of the Potential Well’s Topology
On the Fractional NLS Equation and the Effects of the Potential Well’s Topology
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分数阶NLS方程及势井拓扑的影响
DOI：
10.1515/ans-2020-2114
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发表时间：
2020
影响因子：
1.8
通讯作者：
Marco Gallo
中科院分区：
文献类型：
--
作者：
S. Cingolani;Marco Gallo
本文考虑分数阶非线性薛定谔方程ε2S(-Δ)Sv+V(X)v=f(V)，x∈ℝN，varepsilon^{2s}(-\Delta)^{S}v+V(X)v=f(V)，其中S∈(0，1){S\in(0，1)}，N≥2{N\geq 2}，F是满足Berestycki-Lions型条件的非线性项，V∈C(ℝN，ℝ){V\in C(\mathbb{R}^{N}，\mathbb{R})}是正势。对于ε>；0{varepsilon>；0}小，我们证明了至少存在Cul(K)+1{{rm cul}(K)+1}正解，其中K是有界势井中的一组局部极小，cul(K){{rm cul}(K)}表示K的杯长，我们用变分方法分析了不定泛函的两个能级在期望解的邻域内的拓扑差异.由于非局域性直接出现在空间的分解中，我们通过适当的半范数引入了一个新的分数质心。其他一些微妙的方面严格地与非本地操作员的存在有关。利用基于分数阶De Giorgi类的正则性结果，我们证明了对于εSmall，所找到的解按多项式衰减且集中在K的某个点附近。
Abstract In this paper we consider the fractional nonlinear Schrödinger equation ε 2 s ( - Δ ) s v + V ( x ) v = f ( v ) , x ∈ ℝ N , \varepsilon^{2s}(-\Delta)^{s}v+V(x)v=f(v),\quad x\in\mathbb{R}^{N}, where s∈(0,1){s\in(0,1)}, N≥2{N\geq 2}, f is a nonlinearity satisfying Berestycki–Lions type conditions and V∈C(ℝN,ℝ){V\in C(\mathbb{R}^{N},\mathbb{R})} is a positive potential. For ε>0{\varepsilon>0} small, we prove the existence of at least cupl(K)+1{{\rm cupl}(K)+1} positive solutions, where K is a set of local minima in a bounded potential well and cupl(K){{\rm 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.