# Multiplicity and concentration of solutions for fractional Schr\"odinger systems via penalization method

## TL;DR

This paper establishes the existence, multiplicity, and concentration of positive solutions for a system of fractional Schrödinger equations using penalization, variational methods, and topological tools, highlighting the influence of potential minima.

## Contribution

It introduces a novel approach combining penalization and topological methods to analyze fractional Schrödinger systems with variable potentials.

## Key findings

- Multiple positive solutions exist depending on the topology of potential minima.
- Solutions concentrate near the minima of the potentials as the parameter tends to zero.
- The number of solutions relates to the topological complexity of the potential's minimum set.

## Abstract

The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}, \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v) &\mbox{ in } \mathbb{R}^{N}, u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth. In order to relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values, we apply penalization techniques, Nehari manifold arguments and Ljusternik-Schnirelmann theory.