Virial functional and dynamics for nonlinear Schrödinger equations of local interactions

  • Takafumi Akahori
  • Hiroaki Kikuchi
  • Takeshi Yamada


Our aim is to verify that the functional in the virial identity classifies the dynamics for nonlinear Schrödinger equations of local interactions. In particular, we give a condition under that there exist stable ground states. Our proof of this stability result is based on the ideas in Colin (Ann Inst H Pincaré 23:753–764, 2006) and Shatah (Math Phys 91:313–327, 1983). However, we emphasize that our argument does not use the strict convexity of the \(\dot{H}^{1}\)-norm of ground state with respect to \(\omega \): a key lemma is Lemma 4.8 below. Furthermore, we discuss the limiting profile of ground states (see Theorem 4.4).


Virial functional Ground state Stability Scattering Blowup Variational method Limiting profile 

Mathematics Subject Classification

35J20 35Q55 37K40 37K45 


  1. 1.
    Akahori, T., Nawa, H.: Blowup and scattering problems for the nonlinear Schrödinger equations. Kyoto J. Math. 53, 629–672 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akahori, T., Kikuchi, H., Nawa, H.: Scattering and blowup problems for a class of nonlinear Schrödinger equations. Differ. Integral Equ. 25, 1075–1118 (2012)MATHGoogle Scholar
  3. 3.
    Akahori, T., Ibrahim, S., Kikuchi, H., Nawa, H.: Global dynamics above the ground state energy for the combined power type nonlinear Schrödinger equations with energy critical growth at low frequencies, preprint, arXiv: 1510.08034
  4. 4.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)MathSciNetMATHGoogle Scholar
  5. 5.
    Brezis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York (2003)MATHGoogle Scholar
  7. 7.
    Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^{s}\). Nonlin. Anal. 14, 807–836 (1990)CrossRefMATHGoogle Scholar
  8. 8.
    Colin, M., Ohta, M.: Stability of solitary waves for derivative nonlinear Schrödinger equation. Ann. Inst. H. Poincaré 23, 753–764 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}}^{n}\). Commun. Pure Appl. Math. 45, 1217–1269 (1992)CrossRefMATHGoogle Scholar
  10. 10.
    Duyckaerts, T., Holmer, J., Roudenko, S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15, 1233–1250 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fröhlich, J., Lieb, E.H., Loss, M.: Stability of coulomb systems with magnetic fields I. The one-electron atom. Commun. Math. Phys. 78, 160–190 (1989)MATHGoogle Scholar
  12. 12.
    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^{N}\). Math. Anal. Appl. 7, 369–402 (1981)MATHGoogle Scholar
  13. 13.
    Glassey, R.T.: On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18, 1794–1797 (1977)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in \({\mathbb{R}}^{N}\): mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlin. Anal. 35, 253–276 (2010)MATHGoogle Scholar
  15. 15.
    Kato, T.: Nonlinear Schrödinger equations. Schrödinger operators (Sonderborg, 1988), 218–263. Lecture Notes in Physics, 345. Springer, Berlin (1989)Google Scholar
  16. 16.
    Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166, 645–675 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Killip, R., Visan, M.: The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Am. J. Math. 132, 361–424 (2010)CrossRefMATHGoogle Scholar
  18. 18.
    Lieb, E.H.: On the lowest eigenvalue of Laplacian for the intersection of two domains. Invent. Math. 74, 441–448 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nakanishi, K.: Remarks on the energy scattering for nonlinear Klein-Gordon and Schrödinger equations. Tohoku Math. J. 53, 285–303 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pucchi, P., Serrin, J.: Uniqueness of ground states for quasilinear elliptic operators. Indiana Univ. Math. J. 47, 501–528 (1998)MathSciNetMATHGoogle Scholar
  21. 21.
    Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data. N. Y. J. Math. 11, 57–80 (2005)MATHGoogle Scholar
  22. 22.
    Shatah, J.: Stable standing waves of nonlinear Klein-Gordon equations. Commun. Math. Phys. 91, 313–327 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Takafumi Akahori
    • 1
  • Hiroaki Kikuchi
    • 2
  • Takeshi Yamada
    • 3
  1. 1.Shizuoka UniversityHamamatsuJapan
  2. 2.Tsuda UniversityKodairaJapan
  3. 3.NagoyaJapan

Personalised recommendations