*Published Paper*

**Inserted:** 10 may 2022

**Last Updated:** 31 may 2023

**Journal:** Advances in Differential Equations

**Volume:** 26

**Number:** 9-10

**Pages:** 397-424

**Year:** 2021

**Abstract:**

Goal of this paper is to study positive semiclassical solutions of the
nonlinear Schr\"odinger equation $$ \varepsilon^{{2s}}(- \Delta)^{s} u+ V(x) u=
f(u), \quad x \in \mathbb{R}^{N,$$} where $s \in (0,1)$, $N \geq 2$, $V \in
C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical
and satisfying general Berestycki-Lions type conditions. We obtain existence
and multiplicity for $\varepsilon>0$ small, where the number of solutions is
related to the cup-length of a set of local minima of $V$. Furthermore, these
solutions are proved to concentrate in the potential well, exhibiting a
polynomial decay. We highlight that these results are new also in the limiting
local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.