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Ground state solutions for a Schrödinger-Poisson system with unconventional potential

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Abstract

We consider the Schrödinger-Poisson system with nonlinear term Q(x)|u|p−1u, where the value of \(\mathop {\lim}\limits_{\left| x \right| \to \infty} \,\,Q\left(x \right)\)Q(x) may not exist and Q may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks.

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References

  1. Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2):283–293

    Article  MathSciNet  Google Scholar 

  2. Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev Math Phys, 2002, 14(4):409–420

    Article  MathSciNet  Google Scholar 

  3. Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76(1):257–274

    Article  MathSciNet  Google Scholar 

  4. Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10(3):391–404

    Article  MathSciNet  Google Scholar 

  5. Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schröodinger-Maxwell equations. J Math Anal Appl, 2008, 345(1):90–108

    Article  MathSciNet  Google Scholar 

  6. Azzollini A, d’Avenia P, Pomponio A. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(2):779–791

    Article  MathSciNet  Google Scholar 

  7. d’Avenia P. Non-radially symmetric solutions of nonlinear Schröodinger equation coupled with Maxwell equations. Adv Nonlinear Stud, 2002, 2(2):177–192

    Article  MathSciNet  Google Scholar 

  8. D’Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3):307–322

    Article  MathSciNet  Google Scholar 

  9. D’Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schröodinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134(5):893–906

    Article  MathSciNet  Google Scholar 

  10. Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2):655–674

    Article  MathSciNet  Google Scholar 

  11. Jiang Y S, Zhou H S. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251(3):582–608

    Article  MathSciNet  Google Scholar 

  12. Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for Schroödinger-Pisson equations with steep well potential. J Differential Equations, 2013, 255(1):1–23

    Article  MathSciNet  Google Scholar 

  13. Zhao L G, Zhao F K. On the existence of solutions for the Schröodinger-Poisson equations. J Math Anal Appl, 2008, 346(1):155–169

    Article  MathSciNet  Google Scholar 

  14. Cerami G, Vaira G. Positive solutions for some non-autonomous Schrödinger-Poisson systems. J Differential Equations, 2010, 248(3):521–543

    Article  MathSciNet  Google Scholar 

  15. Vaira G. Ground states for Schroödinger-Poisson type systems. Ric Mat, 2011, 60(2):263–297

    Article  MathSciNet  Google Scholar 

  16. Li G B, Peng S J, Yan S S. Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system. Commun Contemp Math, 2010, 12(6):1069–1092

    Article  MathSciNet  Google Scholar 

  17. Batista A M, Furtado M F. Positive and nodal solutions for a nonlinear Schrödinger-Poisson system with sign-changing potentials. Nonlinear Anal Real World Appl, 2018, 39: 142–156

    Article  MathSciNet  Google Scholar 

  18. Cerami G, Molle R. Positive bound state solutions for some Schrödinger-Poisson systems. Nonlinearity, 2016, 29(10):3103–3119

    Article  MathSciNet  Google Scholar 

  19. Chen J Q. Multiple positive solutions of a class of non autonomous Schrödinger-Poisson systems. Nonlinear Anal Real World Appl, 2015, 21: 13–26

    Article  MathSciNet  Google Scholar 

  20. Huang L R, Rocha E M, Chen J Q. Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity. J Differential Equations, 2013, 255(8):2463–2483

    Article  MathSciNet  Google Scholar 

  21. Wang C H, Yang J. Positive solution for a nonlinear Schrödinger-Poisson system. Discrete Contin Dyn Syst, 2018, 38(11):5461–5504

    Article  MathSciNet  Google Scholar 

  22. Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71(10):4927–4934

    Article  MathSciNet  Google Scholar 

  23. Zhong X J, Tang C L. Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in ℝ3. Nonlinear Anal Real World Appl, 2018, 39: 166–184

    Article  MathSciNet  Google Scholar 

  24. Ye Y W, Tang C L. Existence and multiplicity results for the Schrödinger-Poisson system with superlinear or sublinear terms. Acta Math Sci, 2015, 35A: 668–682

    MATH  Google Scholar 

  25. Ruiz D. Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere. Math Models Methods Appl Sci, 2005, 15(1):141–164

    Article  MathSciNet  Google Scholar 

  26. Ianni I, Vaira G. On concentration of positive bound states for the Schrödinger-Poisson problem with potentials. Adv Nonlinear Stud, 2008, 8(3):573–595

    Article  MathSciNet  Google Scholar 

  27. He Y, Li G B. Standing waves for a class of Schrödinger-Poisson equations in ℝ3 involving critical Sobolev exponents. Ann Acad Sci Fenn Math, 2015, 40(2):729–766

    Article  MathSciNet  Google Scholar 

  28. Kwong M K. Uniqueness of positive solutions of Δu − u + up = 0 in ℝN. Arch Rational Mech Anal, 1989, 105(3):243–266

    Article  MathSciNet  Google Scholar 

  29. Berestycki H, Lions P L. Nonlinear scalar field equations. I. Existence of a ground state. Arch Rational Mech Anal, 1983, 82(4):313–345

    Article  MathSciNet  Google Scholar 

  30. Yang J F, Yu X H. Existence of solutions for a semilinear elliptic equation in ℝN with sign-changing weight. Adv Nonlinear Stud, 2008, 8(2):401–412

    Article  MathSciNet  Google Scholar 

  31. Willem M. Minimax Theorems. Boston: Birkhäuser, 1996

    Book  Google Scholar 

  32. Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88(3): 486–490

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Yao Du & Chunlei Tang

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Yao Du

Authors
  1. Yao Du
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  2. Chunlei Tang
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Corresponding author

Correspondence to Chunlei Tang.

Additional information

The second author was supported by National Natural Science Foundation of China (11471267) and the first author is supported by Graduate Student Scientific Research Innovation Projects of Chongqing (CYS17084).

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Du, Y., Tang, C. Ground state solutions for a Schrödinger-Poisson system with unconventional potential. Acta Math Sci 40, 934–944 (2020). https://doi.org/10.1007/s10473-020-0404-2

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  • DOI: https://doi.org/10.1007/s10473-020-0404-2

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