Abstract
In this paper, we study a nonlinear Schrödinger–Poisson system
where \({\mu > 0}\) is a parameter, \({V_{\lambda }}\) is allowed to be sign-changing and f is an indefinite function. We require that \({V_{\lambda }:=\lambda V^{+}-V^{-}}\) with V + having a bounded potential well Ω whose depth is controlled by λ and \({V^{-} \geq 0}\) for all \({x\in \mathbb{R} ^{3}}\). Under some suitable assumptions on K and f, the existence and the nonexistence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is explored as well.
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J. Sun was supported by the National Natural Science Foundation of China (Grant Nos. 11201270 and 11271372), Shandong Natural Science Foundation (Grant No. ZR2012AQ010), Young Teacher Support Program of Shandong University of Technology, and China Postdoctoral Science Foundation (Grant No. 2014M551494). T.F. Wu was supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan.
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Sun, J., Wu, Tf. On the nonlinear Schrödinger–Poisson systems with sign-changing potential. Z. Angew. Math. Phys. 66, 1649–1669 (2015). https://doi.org/10.1007/s00033-015-0494-1
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DOI: https://doi.org/10.1007/s00033-015-0494-1