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

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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 3/4 < s < 1, 2* s := 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

Acknowledgments

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.

References

  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: 1–19CrossRefGoogle Scholar
  2. 2.
    Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76: 257–274MathSciNetCrossRefMATHGoogle 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: 79–93MathSciNetCrossRefMATHGoogle 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: 29–48MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Benci V, Cerami G, Passaseo D. On the number of positive solutions of some nonlinear elliptic problems. Nonlinear Anal, 1991, 114: 93–107MathSciNetMATHGoogle Scholar
  6. 6.
    Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11: 283–293MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev Math Phys, 2002, 14: 409–420MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Caffarelli L, Roquejoffer J, Savin O. Nonlocal minimal surfaces. Comm Pure Appl Math, 2010, 63: 1111–1144MathSciNetMATHGoogle Scholar
  10. 10.
    Chen G Y. Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations. Nonlinearity, 2015, 28: 927–949MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen G Y, Zheng Y Q. Concentration phenomenon for fractional nonlinear Schrödinger equations. Comm Pure Appl Anal, 2014, 13: 2359–2376CrossRefMATHGoogle Scholar
  12. 12.
    Cont R, Tankov P. Financial Modeling with Jump Processes. Boca Raton: Chapman Hall/CRC Financial Mathematics Series, 2004MATHGoogle Scholar
  13. 13.
    Dávila J, det Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrodinger equation. J Differential Equations, 2014, 256: 858–892MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional sobolev spaces. Bull Sci Math, 2012, 136: 521–573MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47: 324–353MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fall M, Mahmoudi F, Valdinoci E. Ground states and concentration phenomena for fractional Schrödinger equation. Nonlinearity, 2015, 28: 1937–1961MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    He X M. Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations. Z Angew Math Phys, 2011, 62: 869–889MathSciNetCrossRefMATHGoogle 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: 023702MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ianni I. Solutions of the Schrödinger-Poisson problem concentrating on spheres, II: Existence. Math Models Methods Appl Sci, 2009, 19: 877–910MathSciNetCrossRefMATHGoogle 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: 573–595MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jeanjean L, Tanaka K. A positive solution for a nonlinear Schrödinger equation on RN. Indiana Univ Math J, 2005, 54: 443–464MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251: 582–608MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Laskin N. Fractals and quantum mechanics. Chaos, 2000, 10: 780–790MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268: 298–305MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lions P L. Solutions of Hartree-Fock equations for Coulomb systems. Comm Math Phys, 1984, 109: 33–97MathSciNetCrossRefMATHGoogle 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: 161–208MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Molica Bisci G, Radulescu V. Ground state solutions of scalar field fractional Schrödinger equations. Calc Var Partial Differential Equations, 2015, 54: 2985–3008MathSciNetCrossRefMATHGoogle 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: 231–258MathSciNetMATHGoogle 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: 799–829MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237: 655–674MathSciNetCrossRefMATHGoogle 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: 253–271MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Secchi S. Ground state solutions for nonlinear Schrödinger equations in R3. J Math Phys, 2013, 54: 031501MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367: 67–102MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Shang X D, Zhang J H. Ground states for fractional Schrödinger equations with critical growth. Nonlinearity, 2014, 27: 187–207MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60: 67–112MathSciNetCrossRefMATHGoogle 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: 3061–3106MathSciNetCrossRefMATHGoogle 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: 809–816MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Willem M. Minimax Theorems. Boston: Birkhäuser, 1996CrossRefMATHGoogle 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: 15–30MathSciNetMATHGoogle 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: 1–23MathSciNetCrossRefMATHGoogle 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: 155–169MathSciNetCrossRefMATHGoogle 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

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