Complete classification of ground state solutions with different Morse index for critical fractional Laplacian system

In this paper, we first obtain the existence of positive ground state solutions for the following critical fractional Laplacian system:( − Δ ) s u = μ 1 | u |2 s ∗ − 2 u + α γ2 s ∗| u | α − 2 u | v | βinℝ n ,( − Δ ) s v = μ 2 | v |2 s ∗ − 2 v + β γ2 s ∗| u | α | v | β − 2 vinℝ n ,then we give a complete classification of positive ground state solutions with different Morse index. More precisely, we show that if ( u ,  v ) be any positive ground state solution of system (1.1), then ( u ,  v ) must be ( C 1 U ϵ ,  y ,  C 2 U ϵ ,  y ) type with Morse index 1 and Morse index 2, where U ϵ ,  y is a positive ground state solution for a given equation.