Partially concentrating standing waves for weakly coupled Schrödinger systems

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations (Formula presented.) where V1 and V2 are radial potentials bounded from below. We address the case, m2 constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.