论文标题
在分数NLS方程以及潜在井拓扑的影响
On the fractional NLS equation and the effects of the potential well's topology
论文作者
论文摘要
在本文中，我们考虑了分数非线性schrödinger方程$$ \ varepsilon^{2S}（ - δ） c（\ mathbb {r}^n，\ mathbb {r}）$是一个积极的潜力，$ f $是满足Berestycki-Lions类型条件的非线性。对于$ \ varepsilon> 0 $ smill，我们证明至少存在$ \ rm {cupl}（k）+1 $阳性解决方案，其中$ k $是一组本地最小值，中有界的潜在井，$ \ rm {cupl}（cupl}（k）$表示$ k $的杯状长度。通过各种方法，我们分析了预期溶液附近的两个不确定功能之间的拓扑差异。由于非局部性直接出现在空间的分解中，因此我们通过合适的半径引入了一个新的质量分数中心。其他一些微妙的方面与非本地操作员的存在严格有关。通过使用基于分数De Giorgi类的规律性结果，我们表明找到的解决方案是多项式衰减的，并集中在$ k $的某个点$ \ varepsilon $ small的某个点。
In this paper we consider the fractional nonlinear Schrödinger equation $$\varepsilon^{2s}(-Δ)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N$$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is a nonlinearity satisfying Berestycki-Lions type conditions. For $\varepsilon>0$ small, we prove the existence of at least $\rm{cupl}(K)+1$ positive solutions, where $K$ is a set of local minima in a bounded potential well and $\rm{cupl}(K)$ 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 $\varepsilon$ small.
