*preprint*

**Inserted:** 10 may 2022

**Year:** 2020

**Abstract:**

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.