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Science China Mathematics

, Volume 61, Issue 6, pp 1039–1062 | Cite as

The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth

  • Yuanyang Yu
  • Fukun Zhao
  • Leiga Zhao
Articles
  • 83 Downloads

Abstract

In this paper, we study the existence and multiplicity of solutions for the following fractional Schrödinger-Poisson system:
$$\left\{ \begin{gathered} {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}u + V\left( x \right)u + \phi u = {\left| u \right|^{2_s^* - 2}}u + f\left( u \right)in{\mathbb{R}^3}, \hfill \\ {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}\phi = {u^2}in{\mathbb{R}^3}, \hfill \\ \end{gathered} \right.$$
(0.1)
where \(\frac{3}{4} < s < 1, 2_s^*:=\frac{6}{3-2s}\) the fractional critical exponent for 3-dimension, the potential V: ℝ3 → ℝ is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System (0.1) via variational methods.

Keywords

fractional Schrödinger-Poisson system critical growth variational methods 

MSC(2010)

35R11 35J50 35B40 35Q40 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11361078 and 11661083). The authors thank the unknown referees for their careful reading and suggestions which improved this work.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsYunnan Normal UniversityKunmingChina
  2. 2.Department of MathematicsBeijing University of Chemical TechnologyBeijingChina

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