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On ground states for the Schrödinger-Poisson system with periodic potentials

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Abstract

This paper is concerned with the following Schrödinger-Poisson system

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + V\left( x \right)u - K\left( x \right)\phi \left( x \right)u = q\left( x \right){{\left| u \right|}^{p - 2}}u,}&{in\;{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2},}&{in\;{\mathbb{R}^3},} \end{array}} \right.$$

where p ∈ (2, 6), V(x) ∈ C(ℝ3, ℝ) is a general periodic function, K(x) and q(x) are nonperiodic functions. Under suitable assumptions, we prove the existence of ground state solutions via variational methods for strongly indefinite problems.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205, Hunan, P. R. China

    Wen Zhang, Jian Zhang & Xiaoliang Xie

  2. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, P. R. China

    Wen Zhang

Authors
  1. Wen Zhang
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  2. Jian Zhang
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  3. Xiaoliang Xie
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Correspondence to Jian Zhang.

Additional information

This work is partially supported by the NNSF (Nos. 11571370, 11471137, 11471278, 61472136).

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Zhang, W., Zhang, J. & Xie, X. On ground states for the Schrödinger-Poisson system with periodic potentials. Indian J Pure Appl Math 47, 449–470 (2016). https://doi.org/10.1007/s13226-016-0177-4

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  • DOI: https://doi.org/10.1007/s13226-016-0177-4

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