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Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term

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Abstract

The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.

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Acknowledgements

We would like to thank the anonymous reviewer’s helpful comments, which greatly improve the presentation of our manuscript. This work was done when the first author visited Department of Mathematics, College of William and Mary during the academic year 2015–2016, and she would like to thank Department of Mathematics, College of William and Mary for their support and kind hospitality.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Yuhua Li & Fuyi Li

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

Authors
  1. Yuhua Li
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  2. Fuyi Li
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  3. Junping Shi
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Corresponding author

Correspondence to Junping Shi.

Additional information

Communicated by A. Malchiodi.

Partially supported by National Natural Science Foundation of China (Grant No. 11301313, 11671239, 11101250), and Science Council of Shanxi Province (2014021009-1, 2015021007) and Shanxi 100 Talent program.

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Li, Y., Li, F. & Shi, J. Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term. Calc. Var. 56, 134 (2017). https://doi.org/10.1007/s00526-017-1229-2

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  • DOI: https://doi.org/10.1007/s00526-017-1229-2

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