# American Institute of Mathematical Sciences

January  2019, 18(1): 237-253. doi: 10.3934/cpaa.2019013

## Critical system involving fractional Laplacian

 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received  October 2017 Revised  May 2018 Published  August 2018

Fund Project: The authors were supported by NSFC grant 11571125.

In this paper, we study the following critical system with fractional Laplacian:
 $\begin{equation*}\begin{cases}(-Δ)^{s}u = μ_{1}|u|^{2^{*}-2}u+\dfrac{αγ}{2^{*}}|u|^{α-2}u|v|^{β} \ \ \ \text{in} \ \ \mathbb{R}^{n},\\(-Δ)^{s}v = μ_{2}|v|^{2^{*}-2}v+\dfrac{βγ}{2^{*}}|u|^{α}|v|^{β-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n},\\u,v∈ D_{s}(\mathbb{R}^{n}).\end{cases}\end{equation*}$
By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.
Citation: Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
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