Semiclassical states for coupled nonlinear Schrodinger equations with critical frequency

In this paper, we are concerned with the coupled nonlinear Schrodinger system [-epsilon(2) Delta u + a(x)u = mu(1)u(3) + beta upsilon(2)u in R-N, -epsilon(2) Delta upsilon + b(x)upsilon = mu(2)upsilon(3) + beta u(2)upsilon in R-N, where 1 <= N <= 3, mu(1), mu(2), beta > 0, a(x) and b(x) are nonnegative continuous potentials, and epsilon > 0 is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as epsilon -> 0, when a(x) and b(x) achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.