Skip to main content
SpringerLink
Log in

Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Standing wave solutions of coupled nonlinear Hartree equations with nonlocal interaction are considered. Such systems arises from mathematical models in Bose–Einstein condensates theory and nonlinear optics. The existence and non-existence of positive ground state solutions are proved under optimal conditions on parameters, and various qualitative properties of ground state solutions are shown. The uniqueness of the positive solution or the positive ground state solution are also obtained in some cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, S., Ogura, A.: Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response. Phys. Rev. E 57(5), 6066–6070 (1998)

    Article  Google Scholar 

  2. Adams, R.-A., Fournier, J.J.F.: Sobolev spaces. In: Pure and Applied Mathematics, 2nd edn, vol. 140. Elsevier, Amsterdam (2003)

  3. Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269(5221), 198–201 (1995)

    Article  Google Scholar 

  5. Bartsch, T., Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37(3–4), 345–361 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bartsch, T., Wang, Z.-Q., Wei, J.-C.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2(2), 353–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163(1), 41–56 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chang, K.-C.: Methods in Nonlinear Analysis. Springer Monographs in Mathematics. Springer, Berlin (2005)

    Google Scholar 

  9. Chen, W.-X., Li, C.-M., Ou, B.: Classification of solutions for a system of integral equations. Commun. Partial Differ. Equ. 30(1–3), 59–65 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, W.-X., Li, C.-M., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, Z.-J., Zou, W.-M.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205(2), 515–551 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, Z.-J., Zou, W.-M.: An optimal constant for the existence of least energy solutions of a coupled Schrödinger system. Calc. Var. Partial Differ. Equ. 48(3–4), 695–711 (2013)

    Article  MATH  Google Scholar 

  13. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463–512 (1999)

    Article  Google Scholar 

  14. de Figueiredo, D.G., Lopes, O.: Solitary waves for some nonlinear Schrödinger systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(1), 149–161 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Esry, B., Greene, C., Burke, J., Bohn, J.: Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997)

    Article  Google Scholar 

  16. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On a classical limit of quantum theory and the non-linear Hartree equation. Geom. Funct. Anal. (Special Volume, Part I):57–78 (2000). GAFA 2000 (Tel Aviv, 1999)

  17. Fröhlich, J., Tsai, T.-P., Yau, H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225(2), 223–274 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Georgiev, V., Venkov, G.: Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential. J. Differ. Equ. 251(2), 420–438 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ginibre, J., Velo, G.: The classical field limit of scattering theory for nonrelativistic many-boson systems. I. Commun. Math. Phys. 66(1), 37–76 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)

    Article  MathSciNet  Google Scholar 

  21. Ikoma, N.: Uniqueness of positive solutions for a nonlinear elliptic system. NoDEA Nonlinear Differ. Equ. Appl. 16(5), 555–567 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ikoma, N., Tanaka, K.: A local mountain pass type result for a system of nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 40(3–4), 449–480 (2011)

    Article  MATH  Google Scholar 

  23. Krolikowski, W., Bang, O., Nikolov, N.I., Neshev, D., Wyller, J., Rasmussen, J.J., Edmundson, D.: Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. J. Opt. B Quantum Semiclass. Opt. 6(5), S288–S294 (2004)

    Article  Google Scholar 

  24. Le Bris, C., Lions, P.-L.: From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. 42(3), 291–363 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lenzmann, E.: Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2(1), 1–27 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, C.-M., Ma, L.: Uniqueness of positive bound states to Schrödinger systems with critical exponents. SIAM J. Math. Anal. 40(3), 1049–1057 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lieb, E.-H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2):93–105 (1976/77)

  28. Lieb, E.-H., Loss, M.: Analysis, vol. 14. In: Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001)

  29. Lieb, E.-H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53(3), 185–194 (1977)

    Article  MathSciNet  Google Scholar 

  30. Lin, T.-C., Wei, J.-C.: Ground state of \(N\) coupled nonlinear Schrödinger equations in \({ R}^n\), \(n\le 3\). Commun. Math. Phys. 255(3), 629–653 (2005)

    Article  MATH  Google Scholar 

  31. Lin, T.-C., Wei, J.-C.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 403–439 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lions, P.-L.: Some remarks on Hartree equation. Nonlinear Anal. 5(11), 1245–1256 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lions, P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109(1), 33–97 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Liu, Z.-L., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282(3), 721–731 (2008)

    Article  MATH  Google Scholar 

  36. Liu, Z.-L., Wang, Z.-Q.: Ground states and bound states of a nonlinear Schrödinger system. Adv. Nonlinear Stud. 10(1), 175–193 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Maia, L., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)

    Article  MATH  Google Scholar 

  39. Mandel, R.: Minimal energy solutions for repulsive nonlinear Schrödinger systems. J. Differ. Equ. 257(2), 450–468 (2014)

    Article  MATH  Google Scholar 

  40. Mitchell, M., Chen, Z.-G., Shih, M.-F., Segev, M.: Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996)

    Article  Google Scholar 

  41. Mitchell, M., Segev, M.: Self-trapping of incoherent white light. Nature 387(6636), 880–883 (1997)

    Article  Google Scholar 

  42. Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. J. Eur. Math. Soc. (JEMS) 10(1), 47–71 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ni, W.-M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  44. Noris, B., Tavares, H., Terracini, S., Verzini, G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267–302 (2010)

    Article  MATH  Google Scholar 

  45. Pedri, P., Santos, L.: Two-dimensional bright solitons in dipolar Bose–Einstein condensates. Phys. Rev. Lett. 95, 200404 (2005)

    Article  Google Scholar 

  46. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28(5), 581–600 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  47. Penrose, R.: The Road to Reality: A Complete Guide to the Laws of the Universe. Alfred A. Knopf Inc., New York (2005)

    MATH  Google Scholar 

  48. Pomponio, A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 227(1), 258–281 (2006)

    Article  MATH  Google Scholar 

  49. Santos, L., Shlyapnikov, G.V., Zoller, P., Lewenstein, M.: Bose–Einstein condensation in trapped dipolar gases. Phys. Rev. Lett. 85, 1791–1794 (2000)

    Article  Google Scholar 

  50. Sato, Y., Wang, Z.Q.: On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(1), 1–22 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  51. Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \(\mathbb{R}^n\). Commun. Math. Phys. 271(1), 199–221 (2007)

    Article  MATH  Google Scholar 

  52. Terracini, S., Verzini, G.: Multipulse phases in \(k\)-mixtures of Bose–Einstein condensates. Arch. Ration. Mech. Anal. 194(3), 717–741 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. Timmermans, E.: Phase separation of Bose–Einstein condensates. Phys. Rev. Lett. 81, 5718–5721 (1998)

    Article  Google Scholar 

  54. Tod, P., Moroz, I.M.: An analytical approach to the Schrödinger–Newton equations. Nonlinearity 12(2), 201–216 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  55. Vaira, G.: Existence of bound states for Schrödinger–Newton type systems. Adv. Nonlinear Stud. 13(2), 495–516 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wang, J., Shi, J.-P.: Standing waves of a weakly coupled Schrödinger system with distinct potential functions. J. Differ. Equ. 260(2), 1830–1864 (2016)

    Article  MATH  Google Scholar 

  57. Wei, J.-C., Weth, T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21(2), 305–317 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  58. Wei, J.-C., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190(1), 83–106 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  59. Wei, J.-C., Winter, M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50(1), 012905,22 (2009)

    Article  MATH  Google Scholar 

  60. Wei, J.-C., Yao, W.: Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 11(3), 1003–1011 (2012)

    MATH  MathSciNet  Google Scholar 

  61. Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston Inc., Boston (1996)

  62. Yang, M.-B., Wei, Y.-H., Ding, Y.-H.: Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities. Z. Angew. Math. Phys. 65(1), 41–68 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

J. Wang was supported by NNSFC (Grants 11571140, 11671077, 11371090), Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Six big talent peaks project in Jiangsu Province (XYDXX-015), and NSF of Jiangsu Province (BK20150478).

Author information

Authors and Affiliations

  1. Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Jun Wang

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

Authors
  1. Jun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junping Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junping Shi.

Additional information

Communicated by M. Struwe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Shi, J. Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction. Calc. Var. 56, 168 (2017). https://doi.org/10.1007/s00526-017-1268-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1268-8

Mathematics Subject Classification

Search

Navigation