Critical system involving fractional Laplacian

Maoding Zhen; Jinchun He; Haoyun Xu

doi:10.3934/cpaa.2019013

Article Contents

2019, Volume 18, Issue 1: 237-253. Doi: 10.3934/cpaa.2019013

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Critical system involving fractional Laplacian

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

Received: September 30, 2017

Revised: April 30, 2018

Published: August 2018

The authors were supported by NSFC grant 11571125.

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J50, 35B33, 35R11.

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