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 {(-Delta)(s)u=mu(1)vertical bar u vertical bar(2s)*(-2)u+alpha gamma/2(s)*vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta) in Rn, (-Delta)(s)v=mu(2)vertical bar v vertical bar(2s)*(-2)v+beta gamma/2(s)*vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v in R-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 (C1U epsilon, y, C2U epsilon, y) type with Morse index 1 and Morse index 2, where U-epsilon,U- y is a positive ground state solution for a given equation.