# Critical system involving fractional Laplacian

## TL;DR

This paper investigates a critical fractional Laplacian system, establishing conditions for the existence or nonexistence of positive least energy solutions using the Nehari manifold approach.

## Contribution

It introduces new existence and nonexistence results for positive solutions of a fractional Laplacian system with critical nonlinearity, under specific conditions.

## Key findings

- Established existence of positive least energy solutions under certain conditions.
- Proved nonexistence of solutions in some parameter regimes.
- Applied Nehari manifold method to analyze the system.

## Abstract

In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n}, u,v\in D_{s}(\mathbb{R}^{n}). \end{cases} \end{equation*} By using the Nehari\ manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.