Existence of solutions to a class of Schrödinger–Poisson systems with indefinite nonlinearity

Abstract

In this paper, we investigate a class of Schrödinger–Poisson systems with indefinite nonlinearity which is a combination of a linear term with parameter and a superlinear term with parameter . Here, the Poisson equation is the form , where . A concentration compactness lemma is established to overcome the lack of compactness. In order to insure the Nehari manifold , the parameters and must have some restriction that compares with and the variation range of depends on l. By seeking the local minimizer of the energy functional on the Nehari manifold, we obtain the existence of solution for the system.

Keywords:

• Schrödinger–Poisson system
• indefinite weight
• Nehari manifold
• concentration compactness

Acknowledgements

The authors would like to thank the anonymous referee for some valuable comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by National Natural Science Foundation of China [grant number 11301313], [grant number 11571209]; Science Council of Shanxi Province [grant number 2013021001-4], [grant number 2014021009-1], [grant number 2015021007].