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


In this paper, we study the existence and multiplicity of solutions for the following fractional Schrödinger-Poisson system:
$$\left\{ \begin{gathered} {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}u + V\left( x \right)u + \phi u = {\left| u \right|^{2_s^* - 2}}u + f\left( u \right)in{\mathbb{R}^3}, \hfill \\ {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}\phi = {u^2}in{\mathbb{R}^3}, \hfill \\ \end{gathered} \right.$$
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.


fractional Schrödinger-Poisson system critical growth variational methods 


35R11 35J50 35B40 35Q40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. 1.
    Alves C O, Miyagaki O H. Existence and concentration of solution for a class of fractional elliptic equation in ℝN via penalization method. Calc Var Partial Differential Equations, 2016, 55: 1–19MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76: 257–274MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benci V, Cerami G. The effect of domain topology on the number of positive solutions of nonlinear elliptic problems. Arch Ration Mech Anal, 1991, 114: 79–93MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benci V, Cerami G. Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc Var Partial Differential Equations, 1994, 2: 29–48MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benci V, Cerami G, Passaseo D. On the number of positive solutions of some nonlinear elliptic problems. Nonlinear Anal, 1991, 114: 93–107MathSciNetzbMATHGoogle Scholar
  6. 6.
    Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11: 283–293MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev Math Phys, 2002, 14: 409–420MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caffarelli L, Roquejoffer J, Savin O. Nonlocal minimal surfaces. Comm Pure Appl Math, 2010, 63: 1111–1144MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen G Y. Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations. Nonlinearity, 2015, 28: 927–949MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen G Y, Zheng Y Q. Concentration phenomenon for fractional nonlinear Schrödinger equations. Comm Pure Appl Anal, 2014, 13: 2359–2376CrossRefzbMATHGoogle Scholar
  12. 12.
    Cont R, Tankov P. Financial Modeling with Jump Processes. Boca Raton: Chapman Hall/CRC Financial Mathematics Series, 2004zbMATHGoogle Scholar
  13. 13.
    Dávila J, det Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrodinger equation. J Differential Equations, 2014, 256: 858–892MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional sobolev spaces. Bull Sci Math, 2012, 136: 521–573MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47: 324–353MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fall M, Mahmoudi F, Valdinoci E. Ground states and concentration phenomena for fractional Schrödinger equation. Nonlinearity, 2015, 28: 1937–1961MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    He X M. Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations. Z Angew Math Phys, 2011, 62: 869–889MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    He X M, Zou W M. Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J Math Phys, 2012, 53: 023702MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ianni I. Solutions of the Schrödinger-Poisson problem concentrating on spheres, II: Existence. Math Models Methods Appl Sci, 2009, 19: 877–910MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ianni I, Vaira G. On concentration of positive bound states for the Schrödinger-Poisson problem with potentials. Adv Nonlinear Stud, 2008, 8: 573–595MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jeanjean L, Tanaka K. A positive solution for a nonlinear Schrödinger equation on RN. Indiana Univ Math J, 2005, 54: 443–464MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251: 582–608MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Laskin N. Fractals and quantum mechanics. Chaos, 2000, 10: 780–790MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268: 298–305MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lions P L. Solutions of Hartree-Fock equations for Coulomb systems. Comm Math Phys, 1984, 109: 33–97MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Metzler R, Klafter J. The resraurant at the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J Phys A, 2004, 37: 161–208MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Molica Bisci G, Radulescu V. Ground state solutions of scalar field fractional Schrödinger equations. Calc Var Partial Differential Equations, 2015, 54: 2985–3008MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Molica Bisci G, Radulescu V, Servadei R. Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge: Cambridge University Press, 2016Google Scholar
  29. 29.
    Murcia E G, Siciliao G. Positive semiclassical states for a fractional Schrödinger-Poisson system. Differential Integral Equations, 2017, 30: 231–258MathSciNetzbMATHGoogle Scholar
  30. 30.
    Palatucci G, Pisante A. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc Var Partial Differential Equations, 2014, 50: 799–829MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237: 655–674MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ruiz D, Vaira G. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential. Rev Mat Iberoam, 2011, 27: 253–271MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Secchi S. Ground state solutions for nonlinear Schrödinger equations in R3. J Math Phys, 2013, 54: 031501MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367: 67–102MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Shang X D, Zhang J H. Ground states for fractional Schrödinger equations with critical growth. Nonlinearity, 2014, 27: 187–207MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60: 67–112MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Szulkin A, Weth T. The methods of Nehari manifold. In: Handbook of Nonconvex Analysis and Applications. Boston: International Press, 2010, 597–632Google Scholar
  38. 38.
    Teng K M. Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent. J Differential Equations, 2016, 261: 3061–3106MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wang Z P, Zhou H S. Positive solution for a nonlinear stationary Schrödinger-Poisson system in R3. Discrete Contin Dyn Syst, 2007, 18: 809–816MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Willem M. Minimax Theorems. Boston: Birkhäuser, 1996CrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang J J, do Ó J M, Squassina M. Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity. Adv Nonlinear Stud, 2016, 16: 15–30MathSciNetzbMATHGoogle Scholar
  42. 42.
    Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J Differential Equations, 2013, 255: 1–23MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhao L G, Zhao F K. On the existence of solutions for the Schrödinger-Poisson equations. J Math Anal Appl, 2008, 346: 155–169MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations