Elliptic systems involving Schrödinger operators with vanishing potentials

We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators

$ \begin{equation*} \left\{ \begin{array}{cll} -\Delta u+V_{1}(x)u = \lambda\rho_{1}(x)(u+1)^{r}(v+1)^{p}&\mbox{ in }&\mathbb{R}^{N}\\ -\Delta v+V_{2}(x)v = \mu\rho_{2}(x)(u+1)^{q}(v+1)^{s}&\mbox{ in }&\mathbb{R}^{N},\\ u(x),v(x)\to 0& \mbox{ as}&|x|\to\infty \end{array} \right. \end{equation*} $

where $ p,q,r,s\geq0 $, $ V_{i} $ is a nonnegative vanishing potential, and $ \rho_{i} $ has the property $ (\mathrm{H}) $ introduced by Brezis and Kamin [4].As in that celebrated work we will prove that for every $ R> 0 $ there is a solution $ (u_R, v_R) $ 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 $ \rho_{i} $, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.