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

TL;DR

This paper investigates positive semiclassical solutions of fractional and local nonlinear Schrödinger equations with critical growth, establishing existence, multiplicity, and concentration properties related to the potential's local minima.

Contribution

It provides new existence and multiplicity results for solutions of fractional and local NLS equations with critical growth, including concentration behavior and decay rates.

Findings

Solutions concentrate in potential wells with polynomial decay for fractional case.

Number of solutions relates to the cup-length of local minima set.

Results extend to local case with exponential decay, new in the literature.

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.