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# Elliptic systems involving Schrödinger operators with vanishing potentials

• * Corresponding author: Pedro Ubilla

The second author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT. The third author was partially supported by FONDECYT Grant 1181125, 1161635, 1171691

• 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 .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.

Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60; Secondary: 47J30, 35B05.

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