Note on ground states of a nonlinear Schrِdinger system

We give a sufficient condition for the existence of positive radial ground states of the time-independent Schrödinger system { − Δ u 1 + λ u 1 = μ 1 u 1 3 + β 12 u 2 2 u 1 + β 13 u 3 2 u 1 , in R n , − Δ u 2 + λ u 2 = β 12 u 1 2 u 2 + μ 2 u 2 3 + β 23 u 3 2 u 2 , in R n , − Δ u 3 + λ u 3 = β 13 u 1 2 u 3 + β 23 u 2 2 u 3 + μ 3 u 3 3 , in R n , u 1 ( x ) → 0 , u 2 ( x ) → 0 , u 3 ( x ) → 0 , as | x | → ∞ , where n = 1 , 2 , 3 , λ > 0 , μ j > 0 and β i j > 0 ( i < j ) for i , j = 1 , 2 , 3 . And in some special cases, our conditions are also necessary.