On the fractional NLS equation and the effects of the potential well's topology

In this paper we consider the fractional nonlinear Schr\"odinger equation $$\varepsilon^{2s}(-\Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N$$ 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.