Critical system involving fractional Laplacian

Abstract

In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-Δ)^{s}u= μ_{1}|u|^{2^{\ast}-2}u+\frac{αγ}{2^{\ast}}|u|^{α-2}u|v|^β \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-Δ)^{s}v= μ_{2}|v|^{2^{\ast}-2}v+\frac{βγ}{2^{\ast}}|u|^α|v|^{β-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.

24 pages