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Orbital stability of standing waves for some nonlinear Schrödinger equations. (English) Zbl 0513.35007

MSC:
35B35 Stability in context of PDEs
35J60 Nonlinear elliptic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
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[1] Adams, R.A., Clarke, F.H.: Gross’s logarithmic sobolev inequality: a simple proof (preprint) · Zbl 0421.46029
[2] Berestycki, H., Cazenave, T.: To appear
[3] Berestycki, H., Lions, P.L.: Existence d’ondes solitaires dans des probl?mes nonlin?aires du type Klein-Gordon. C. R. Paris287, 503-506 (1978);288, 395-398 (1979) · Zbl 0391.35055
[4] Berestycki, H., Lions, P.L.: Nonlinear scalar fields equations. Parts I and II. Arch. Rat. Mech. Anal. (to appear) · Zbl 0556.35046
[5] Berger, M.S.: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation. J. Funct. Anal.9, 249-261 (1972) · Zbl 0224.35061
[6] Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys.100, 62-93 (1976)
[7] Cazenave, T.: Equations de Schr?dinger non lin?aires. Th?se de 3?me cycle Univ. P. et M. Curie, Paris (1978)
[8] Cazenave, T.: Equations de Schr?dinger non lin?aires en dimension deux. Proc. R. Soc. Edin88, 327-346 (1979) · Zbl 0428.35021
[9] Cazenave, T.: Stable solutions of the logarithmic Schr?dinger equation. Nonlinear Anal. T.M.A. (to appear) · Zbl 0529.35068
[10] Ginibre, J., Velo, G.: On a class of nonlinear Schr?dinger equations. I. The Gauchy problem, general case. J. Funct. Anal.32, 1-32 (1979) · Zbl 0396.35028
[11] Ginibre, J., Velo, G.: On a class of nonlinear Schr?dinger equations. III. Special theories in dimensions 1, 2, and 3. Ann. Inst. Henri Poincar?28, 287-316 (1978) · Zbl 0397.35012
[12] Ginibre, J., Velo, G.: Equation de Schr?dinger non lin?aire avec interaction non locale. C. R. Paris288, 683-685 (1979) · Zbl 0397.35013
[13] Ginibre, J., Velo, G.: On a class of nonlinear Schr?dinger equations with non local interaction. Math. Zeitschr. (to appear) · Zbl 0407.35063
[14] Glassey, R.T.: On the blowing-up of solutions to the Cauchy Problem for nonlinear Schr?dinger equations. J. Math. Phys.18, 1794-1797 (1977) · Zbl 0372.35009
[15] Hartree, D.: The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods. Proc. Camb. Philos. Soc.24, 89-132 (1968)
[16] Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett.15, 1005-1008 (1965)
[17] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math.57, 93-105 (1977) · Zbl 0369.35022
[18] Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys.53, 185-194 (1974)
[19] Lions, P.L.: The Choquard equation and related equations. Nonlinear Anal. T.M.A.4, 1063-1073 (1980) · Zbl 0453.47042
[20] Lions, P.L.: Some remarks on Hartree equation. Nonlinear Anal. T.M.A.5, 1245-1256 (1981) · Zbl 0472.35074
[21] Lions, P.L.: Principe de concentration ? compacit? en calcul des variations. C. R. Paris294, 261-264 (1982) · Zbl 0485.49005
[22] Lions, P.L.: To appear
[23] Lin, J.E., Strauss, W.: Decay and scattering of solutions of a nonlinear Schr?dinger equation. J. Funct. Anal.30, 245-263 (1978) · Zbl 0395.35070
[24] MacLeod, K., Serrin, J.: Personal communication
[25] Nehari, Z.: On a nonlinear differential equation arising in nuclear physics. Proc. R. Irish Acad.62, 117-135 (1963) · Zbl 0124.30204
[26] Pecher, H., Von Wahl, W.: Time dependent nonlinear Schr?dinger equations. Manuscripta Mathematica (to appear) · Zbl 0399.35030
[27] Reeken, M.: Global theorem on bifurcation and its application to the Hartree equation of the Helium atom. J. Math. Phys.11, 2505-2512 (1970)
[28] Ryder, G.: Boundary value problems for a class of nonlinear differential equations. Pac. J. Math.22, 477-503 (1967) · Zbl 0152.28303
[29] Slater, J.C.: A note on Hartree’s method. Phys. Rev.35, 210-211 (1930)
[30] Strauss, W.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149-162 (1977) · Zbl 0356.35028
[31] Strauss, W.: The nonlinear Schr?dinger equation. Proceedings of the Rio conference, August 1977
[32] Stuart, C.A.: Existence theory for the Hartree equation. Arch. Rat. Mech. Anal.51, 60-69 (1973) · Zbl 0287.34032
[33] Stuart, C.A.: An example in nonlinear functional analysis: the Hartree equation. J. Math. Anal. Appl.49, 725-733 (1975) · Zbl 0311.47032
[34] Suydam, B.R.: Self-focusing of very powerful laser beams. U.S. Dept. of Commerce. N.B.S. Special Publication 387
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