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Elliptic systems involving Schrödinger operators with vanishing potentials

  • * Corresponding author: Pedro Ubilla

    * 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

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  • 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.

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

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