# Multiplicity of solutions for fractional Schr\"odinger systems in $\mathbb{R}^{N}$

## TL;DR

This paper studies the existence and multiplicity of positive solutions for a class of fractional Schrödinger systems in \\mathbb{R}^{N}, linking the number of solutions to the topology of the potential's minimum set using variational methods.

## Contribution

It introduces new results on the multiplicity of solutions for fractional Schrödinger systems, considering both subcritical and critical nonlinearities, and relates solutions to the topology of potential minima.

## Key findings

- Multiple positive solutions exist depending on the topology of the potential minima.
- The number of solutions increases with the complexity of the minimum set topology.
- Results apply to both subcritical and critical nonlinear cases.

## Abstract

In this paper we deal with the following nonlocal systems of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v)+\gamma H_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\ \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v)+\gamma H_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} \\ u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are continuous potentials, $Q$ is a homogeneous $C^{2}$-function with subcritical growth, $\gamma\in \{0, 1\}$ and $H(u, v)=\frac{2}{\alpha+\beta}|u|^{\alpha} |v|^{\beta}$ with $\alpha, \beta\geq 1$ such that $\alpha+\beta=2^{*}_{s}$. We investigate the subcritical case $(\gamma=0)$ and the critical case $(\gamma=1)$, and using Ljusternik-Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values.