We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators {equation presented} where p, q, r, s ≥ 0, Vi is a nonnegative vanishing potential, and ρi has the property (H) introduced by Brezis and Kamin [4]. As in that celebrated work we will prove that for every R > 0 there is a solution (uR, vR) defined on the ball of radius R centered at the origin. Then, we will show that this sequence of solutions tends to a bounded solution of the previous system when R tends to infinity. Furthermore, by imposing some restrictions on the powers p, q, r, s without additional hypotheses on the weights ρi, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.
