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 "Equation missing"where V₁ V 1 and V₂ V 2 are radial potentials bounded from below. We address the case "Equation missing", m₂ m 2 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.