We study the concentrated NLS on \({{\mathbf {R}}}^n\), with power non-linearities, driven by the fractional Laplacian, \((-\Delta )^s, s>\frac{n}{2}\). We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space \(H^s({{\mathbf {R}}}^n)\). Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding \(H^s({{\mathbf {R}}}^n)\hookrightarrow L^\infty ({{\mathbf {R}}}^n)\), which provides an alternative calculation and justification of the sharp constants in these inequalities.

我们研究了\({{\mathbf {R}}}^n\)上的集中 NLS ，具有幂非线性，由分数拉普拉斯算子\((-\Delta )^s, s>\frac{n {2}\)。我们在参数的最佳范围内明确构建孤立波，使它们属于自然能量空间\(H^s({{\mathbf {R}}}^n)\)。接下来，我们提供了它们光谱稳定性的完整分类。最后，我们证明了波是非简并的，因此轨道稳定，只要它们是光谱稳定的。顺便说一下，我们的构造表明，集中 NLS 的孤子剖面实际上是 Sobolev 嵌入的精确极小值\(H^s({{\mathbf {R}}}^n)\hookrightarrow L^\infty ({{\ mathbf {R}}}^n)\)，它提供了这些不等式中尖锐常数的替代计算和证明。