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Existence of a nontrivial solution for a strongly indefinite periodic Choquard system

  • Shaowei Chen
  • Liqin Xiao
Article

Abstract

We consider the Choquard system
$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\Delta u+V( x) u+|u|^{p-2}u=\lambda \phi u , &{}\quad \text{ in } \mathbb {R}^{3},\\ -\Delta \phi = u^{2}, &{}\quad \text{ in } \mathbb {R}^{3}. \end{array} \right. \end{aligned}$$
where \(\lambda >0\) is a parameter, \(3<p<6\), \(V\in C( \mathbb {R}^{3}) \) is \(1\)-periodic in \(x_j\) for \(j = 1,2,3\) and 0 is in a spectral gap of the operator \(-\Delta +V\). This system is strongly indefinite, i.e., the operator \(-\Delta +V\) has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities \(|u|^{p-2}u\) and \(\lambda \phi u\). Moreover, the functional corresponding to this system does not satisfy the Palais–Smale condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small \(\lambda >0,\) this system has a nontrivial solution.

Mathematics Subject Classification

35J20 35J60 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions on the manuscript. Shaowei Chen was supported by Science Foundation of Huaqiao University and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY119).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouChina

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