Standing waves for linearly coupled Schrödinger equations with critical exponent

We study the following linearly coupled Schrödinger equations:{−ε2Δu+a(x)u=up+λv,x∈RN,−ε2Δv+b(x)v=v2⁎−1+λu,x∈RN,u,v>0in RN,u(x),v(x)→0as |x|→∞, where N⩾3N⩾3, 2⁎=2NN−2, 10λ>0, we obtain positive solutions of the coupled system for sufficiently small ε>0ε>0, which have concentration phenomenon as ε→0ε→0. It is interesting that we do not need any further assumptions on b(x)b(x).