Multiplicity and concentration results for local and fractional NLS equations with critical growth

Goal of this paper is to study the following singularly perturbed nonlinear Schr\"odinger equation \r\n$$ \varepsilon^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N,$$\r\nwhere $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. \r\nWhen $\eps>0$ is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of $V$; in particular we improve the result in \cite{HeZo}.\r\nFurthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. \r\nFinally, we prove the previous results also in the limiting local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.