Note on ground states of a nonlinear Schrödinger system

We give a sufficient condition for the existence of positive radial ground states of the time-independent Schrödinger system{−Δu1+λu1=μ1u13+β12u22u1+β13u32u1,in Rn,−Δu2+λu2=β12u12u2+μ2u23+β23u32u2,in Rn,−Δu3+λu3=β13u12u3+β23u22u3+μ3u33,in Rn,u1(x)→0,u2(x)→0,u3(x)→0,as |x|→∞, where n=1,2,3n=1,2,3, λ>0λ>0, μj>0μj>0 and βij>0βij>0(i