Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2019, Article ID 8981528, 8 pages
https://doi.org/10.1155/2019/8981528
Research Article

Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

1School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
2School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

Correspondence should be addressed to Anran Li; nc.ude.uxs@narnail

Received 8 June 2019; Accepted 16 July 2019; Published 31 July 2019

Academic Editor: Richard I. Avery

Copyright © 2019 Lizhen Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. Benci and D. Fortunato, “An eigenvalue problem for the Schrödinger-Maxwell equations,” Topological Methods in Nonlinear Analysis, vol. 11, no. 2, pp. 283–293, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  2. T. D'Aprile and D. Mugnai, “Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 134, no. 5, pp. 893–906, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. Ruiz, “The Schrödinger-Poisson equation under the effect of a nonlinear local term,” Journal of Functional Analysis, vol. 237, no. 2, pp. 655–674, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. Mao, L. Yang, A. Qian, and S. Luan, “Existence and concentration of solutions of Schrödinger-Possion system,” Applied Mathematics Letters, vol. 68, pp. 8–12, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  5. L. Zhao and F. Zhao, “Positive solutions for Schrödinger–Poisson equations with a critical exponent,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2150–2164, 2009. View at Publisher · View at Google Scholar
  6. M. Sun, J. Su, and L. Zhao, “Solutions of a Schrödinger-Poisson system with combined nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 442, no. 2, pp. 385–403, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Shao and A. Mao, “Multiplicity of solutions to Schrödinger-Poisson system with concave-convex nonlinearities,” Applied Mathematics Letters, vol. 83, pp. 212–218, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  8. L. Wang, S. Ma, and X. Wang, “On the existence of solutions for nonhomogeneous Schrödinger-Poisson system,” Boundary Value Problems, vol. 2016, no. 1, article no. 76, 2016. View at Publisher · View at Google Scholar
  9. Z. L. Wei, Existence of Infinitely Many Solutions for the Fractional Schrödinger-Maxwell Eqautions, 2015, https://arxiv.org/abs/1508.03088.
  10. K. Teng, “Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent,” Journal of Differential Equations, vol. 261, no. 6, pp. 3061–3106, 2016. View at Publisher · View at Google Scholar
  11. J. Zhang, J. M. do Ó, and M. Squassina, “Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity,” Advanced Nonlinear Studies, vol. 16, no. 1, pp. 15–30, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. C. Duarte and M. A. Souto, “Fractional Schrödinger-Poisson equations with general nonlinearities,” Electronic Journal of Differential Equations, vol. 319, pp. 1–19, 2016. View at Google Scholar
  13. K. Li, “Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations,” Applied Mathematics Letters, vol. 72, pp. 1–9, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Yu, F. Zhao, and L. Zhao, “The existence and multiplicity of solutions of a fractional Schrodinger-Poisson with critical exponent,” Science China Mathematics, vol. 61, no. 6, pp. 1039–1062, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Xiang, B. Zhang, and X. Guo, “Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 120, pp. 299–313, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Q. Q. Li and X. Wu, “A new result on high energy solutions for Schrödinger-Kirchhoff type equations in ,” Applied Mathematics Letters, vol. 30, pp. 24–27, 2014. View at Publisher · View at Google Scholar
  17. E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces,” Bulletin des Sciences Mathématiques, vol. 136, no. 5, pp. 521–573, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. P. Pucci, M. Xiang, and B. Zhang, “Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in ,” Calculus of Variations and Partial Differential Equations, vol. 54, no. 3, pp. 2785–2806, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  19. K. Teng, “Multiple solutions for a class of fractional Schrodinger equations in ,” Nonlinear Analysis: Real World Applications, vol. 21, pp. 76–86, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  20. H. Brézis, J. Coron, and L. Nirenberg, “Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,” Communications on Pure and Applied Mathematics, vol. 33, no. 5, pp. 667–684, 1980. View at Publisher · View at Google Scholar
  21. M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  22. T. Bartsch and M. Willem, “On an elliptic equation with concave and convex nonlinearities,” Proceedings of the American Mathematical Society, vol. 123, no. 11, pp. 3555–3561, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  23. P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 7, no. 9, pp. 981–1012, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Z. Binlin, G. Molica Bisci, and R. Servadei, “Superlinear nonlocal fractional problems with infinitely many solutions,” Nonlinearity, vol. 28, no. 7, pp. 2247–2264, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus