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Indian Journal of Pure and Applied Mathematics

, Volume 47, Issue 3, pp 449–470 | Cite as

On ground states for the Schrödinger-Poisson system with periodic potentials

  • Wen Zhang
  • Jian Zhang
  • Xiaoliang Xie
Article
  • 111 Downloads

Abstract

This paper is concerned with the following Schrödinger-Poisson system
$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + V\left( x \right)u - K\left( x \right)\phi \left( x \right)u = q\left( x \right){{\left| u \right|}^{p - 2}}u,}&{in\;{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2},}&{in\;{\mathbb{R}^3},} \end{array}} \right.$$
where p ∈ (2, 6), V(x) ∈ C(ℝ3, ℝ) is a general periodic function, K(x) and q(x) are nonperiodic functions. Under suitable assumptions, we prove the existence of ground state solutions via variational methods for strongly indefinite problems.

Key words

Schrödinger-Poisson system ground state solutions variational methods strongly indefinite functionals 

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References

  1. 1.
    N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277–320.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423–443.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257–274.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391–404.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Azzollini, P. d’Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779–791.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90–108.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    C. O. Alves, M. A. S. Souto and S. H. M. Soares, Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377 (2011), 584–592.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1–22.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283–293.MathSciNetzbMATHGoogle Scholar
  10. 10.
    V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409–420.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521–543.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Chen and Y. Zheng, Stationary solutions of non-autonomous Maxwell-Dirac systems, J. Differential Equations, 255 (2013), 840–864.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDE, 4 (1996), 121–137.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. D’Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinb. A, 134 (2004), 893–906.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    T. D’Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307–322.MathSciNetzbMATHGoogle Scholar
  16. 16.
    T. D’Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321–342.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y. H. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, J. Differential Equations, 246 (2009), 2829–2848.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007–1032.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Y. H. Ding, Variational methods for strongly indefinite problems, World Scientific Press, 2008.Google Scholar
  20. 20.
    X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869–889.CrossRefzbMATHGoogle Scholar
  21. 21.
    X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    W. N. Huang and X. H. Tang, The existence of infinitely many solutions for the nonlinear Schrödinger-Maxwell equations, Results Math., 65 (2014), 223–234.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 415 (2014), 791–802.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations with critical nonlinearity, Taiwanese J. Math., 18 (2014), 1203–1217.MathSciNetCrossRefGoogle Scholar
  25. 25.
    I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573–595.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    I. Ianni, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part II: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877–910.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707–720.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Z. M. Luo, J. Zhang and W. Zhang, Ground state solutions for diffusion system with superlinear nonlinearity, Elec. J. Quali. Theo. Diff. Equa., 17 (2015), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    D. Mugnai, The Schrödinger-Poisson system with positive potential, Commun. Partial Differ. Equ., 36 (2011), 1099–1117.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259–287.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655–674.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141–164.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoam., 27 (2011), 253–271.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    J. T. Sun, H. B. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differential Equations, 252 (2012), 3365–3380.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802–3822.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math., 18 (2014), 1957–1979.MathSciNetGoogle Scholar
  37. 37.
    X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 361–373.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715–728.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche Mat., 2 (2011), 263–297.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.CrossRefzbMATHGoogle Scholar
  41. 41.
    Z. P. Wang and H. S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Maxwell system in ℝ3, Discrete Contin. Dyn. Syst., 18 (2007), 809–816.MathSciNetCrossRefGoogle Scholar
  42. 42.
    M. B. Yang, Ground state solutions for a periodic Schröinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620–2627.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    L. G. Zhao, H. D. Liu and F. K. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equa., 255 (2013), 1–23.CrossRefzbMATHGoogle Scholar
  44. 44.
    J. Zhang, X. Tang and W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    W. Zhang, X. Tang and J. Zhang, Ground state solutions for a diffusion system, Comput. Math. Appl., 69 (2015), 337–346.MathSciNetCrossRefGoogle Scholar
  46. 46.
    L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155–169.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Indian National Science Academy 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHunan University of CommerceHunanP. R. China
  2. 2.School of Mathematics and StatisticsCentral South UniversityHunanP. R. China

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