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

TL;DR

This paper investigates the fractional nonlinear Schrödinger equation with a potential well, proving multiple positive solutions exist and analyzing their topological and decay properties using variational methods and fractional analysis.

Contribution

It introduces a new fractional center of mass concept and establishes existence and concentration results for solutions considering the potential well's topology.

Findings

Existence of at least cup(K)+1 positive solutions for small 

Solutions decay polynomially and concentrate near local minima of the potential

Development of a variational approach accounting for nonlocal fractional operators

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.