In this paper, we first obtain the existence of positive ground state solutions for the following critical fractional Laplacian system:

$$\left\{\begin{array}{l}{(-\mathrm{\Delta })}^{s}u={\mu }_1|u{|}^{2_s^{\ast }-2}u+\frac{\alpha \gamma }{2_s^{\ast }}|u{|}^{\alpha -2}u|v{|}^{\beta }\text{in}{ℝ}^n,\\ {(-\mathrm{\Delta })}^{s}v={\mu }_2|v{|}^{2_s^{\ast }-2}v+\frac{\beta \gamma }{2_s^{\ast }}|u{|}^{\alpha }|v{|}^{\beta -2}v\text{in}{ℝ}^n,\end{array}\right.$$

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_ϵ, y, C₂U_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.

中文翻译：

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临界分数拉普拉斯系统具有不同摩尔斯指数的基态解的完整分类

在本文中，我们首先获得以下临界分数拉普拉斯系统的正基态解的存在：

$$\left\{\begin{array}{l}{（-\mathrm{\Delta }）}^sü={\mu }_{\mathrm{1个}}|ü{|}^{2_s^{\ast }-2}ü+\frac{\alpha \gamma }{2_s^{\ast }}|ü{|}^{\alpha -2}ü|v{|}^{\beta }\text{在}{ℝ}^{ñ}，\\ {（-\mathrm{\Delta }）}^{s}v={\mu }_2|v{|}^{2_s^{\ast }-2}v+\frac{\beta \gamma }{2_s^{\ast }}|ü{|}^{\alpha }|v{|}^{\beta -2}v\text{在}{ℝ}^{ñ}，\end{array}\right.$$

然后我们给出具有不同摩尔斯指数的正基态解的完整分类。更准确地说，我们证明如果（u，  v）是系统（1.1）的任何正基态解，则（u，  v）必须为（C ₁ U _{ϵ，  y}，  C ₂ U _{ϵ，  y}）类型，且具有Morse类型索引1和Morse指标2，其中ü _{ε，  ÿ}是对于给定的方程的正基态溶液。