# Semiclassical states of a linearly coupled critical fractional Schr\"{o}dinger system

## TL;DR

This paper studies the existence and properties of semiclassical states in a coupled fractional Schr"odinger system, revealing how solutions concentrate and decay as parameters vary, with new results on multiplicity under certain conditions.

## Contribution

It establishes existence, decay, and concentration of positive solutions for a coupled fractional Schr"odinger system, including multiplicity results under specific potential assumptions.

## Key findings

- Existence of positive ground states for small epsilon.
- Decay estimates and concentration behavior of solutions.
- Multiplicity of solutions under additional potential conditions.

## Abstract

This paper focuses on the linearly coupled critical fractional Schr\"{o}dinger system \begin{equation*} \begin{cases} \epsilon^{2s}(-\triangle)^s u +a(x)u=u^p+\lambda v\quad &\text{in}\ \mathbb{R}^N,\\ \epsilon^{2s}(-\triangle)^s v +b(x)v=v^{2_s^*-1}+\lambda u\quad &\text{in}\ \mathbb{R}^N, \end{cases} \end{equation*} where $N>2s,$ $s\in(0,1),$ $p\in(1,2_s^*),$ $\epsilon$ and $\lambda$ are positive parameters, $a,b\in C{(\mathbb{R}^N)}$ are positive potentials, and $(-\triangle)^s$ is the fractional Laplacian operator. Under certain assumptions on $a$ and $\lambda,$ we obtain the existence, decay estimates and concentration property of positive vector ground states for small $\epsilon.$ Furthermore, under an additional assumption on potentials $a$ and $b$, we consider the multiplicity of positive vector solutions for small $\epsilon$, which turn out to have similar decay estimate and concentration property to those of the ground state for small $\epsilon$.