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Stationary solutions for the non-linear Hartree equation with a slowly varying potential

  • Marta Macrì
  • Margherita Nolasco
Article

Abstract

We consider the non-linear Hartree equation with a slowly varying external potential \({V_{\varepsilon}}\) and a short range, attractive two-body interaction W. We prove the existence of stationary solutions which are approximatively given by a superposition of several Hartree solitons with their center of mass positions behaving, at the leading order, as classical particles at rest in the background potential \({V_{\varepsilon}}\) .

Mathematics Subject Classification (2000)

35Q55 (35J10) 

Keywords

Elliptic equation Shadowing lemma 

References

  1. 1.
    Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248(2), 423–443 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambrosetti A., Cingolani S., Badiale M.: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140, 285–300 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Angenent, S.: The shadowing lemma for elliptic PDE. Dynamics of infinite- dimensional systems (Lisbon, 1986), pp. 7–22. NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci. vol. 37. Springer, Berlin (1987)Google Scholar
  4. 4.
    Cingolani S., Nolasco M.: Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinburgh 128 A, 1249–1260 (1998)MathSciNetGoogle Scholar
  5. 5.
    Erdös L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many-body Coulomb system. Adv. Theor. Math. Phys. 5, 1169–1205 (2001)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Floer A., Weinstein A.: Nonspreading wave packets for the cubic schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fröhlich J., Gustafson S., Jonsson B.L.G., Sigal I.M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250, 613–642 (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: The Kernel Condition for Boson Stars (in preparation)Google Scholar
  9. 9.
    Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum bose gases and nonlinear hartree equation. Seminaire equations aux Drives Partielles. Exp. No. XIX, cole Polytech., Palaiseau (2004)Google Scholar
  10. 10.
    Fröhlich J., Tsai T.-P., Yau H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225, 223–274 (2002)zbMATHCrossRefGoogle Scholar
  11. 11.
    Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer, Heidelberg (1983)zbMATHGoogle Scholar
  12. 12.
    Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lieb E.H.: Existence and Uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)MathSciNetGoogle Scholar
  14. 14.
    Lions P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case I. Ann. Inst. Poincaré Anal. Non Linéaire 1, 109–145 (1984)zbMATHGoogle Scholar
  15. 15.
    Lions P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case II. Ann. Inst. Poincaré Anal. Non Linéaire 1, 223–283 (1984)zbMATHGoogle Scholar
  16. 16.
    Oh Y.G.: On positive multi-bump states of nonlinear Schrödinger equation under multiple well potentials. Commun. Math. Phys. 131, 223–253 (1990)zbMATHCrossRefGoogle Scholar
  17. 17.
    Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Weinstein M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di L’AquilaL’AquilaItaly

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