[1] Cycon H L, Froese R G, KirschW, Simon B. Schr¨odinger operators and application to quantum mechanics and global geometry studyed. Texts and Monograghs in Physics. Berlin: Springer-Verlag, 1987
[2] Carles R, Nakamura Y. Nonlinear Schr¨odinger equations with stark potential. Hokkaido Math J, 2004, 33: 719–729
[3] Cazenave T. Semilinear Schr¨odinger equations//Courant Lecture Notes in Mathematics, 10, NYU, CIMS, AMS, 2003
[4] de Bouard A. Nonlinear Schr¨odinger equations with magnetic fields. Differential Integral Equations, 1991, 4: 73–88
[5] Ginibre J, Velo G. On a class of nonlinear Schr¨odinger equations. I. The Cauchy problem, general case. J Funct Anal, 1979, 32: 1–32
[6] Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schr¨odinger equations. J Math Phys, 1977, 18: 1794–1797
[7] G´erard P. Description du defaut de compacite de l’injection de Sobolev. ESAIM Control Optim Calc Var, 1998, 3: 213–233
[8] Hmidi T, Keraani S. Blowup theory for the critical nonlinear Schr¨odinger equations revisited. Int Math Res Not, 2005, 46: 2815–2828
[9] Hmidi T, Keraani S. Remarks on the blowup for the L2-critical nonlinear Schr¨odinger equations. SIAM J Math Anal, 2006, 38: 1035–1047
[10] Kwong M K. Uniqueness of positive solutions of △u−u+up = 0 in Rn. Arch Rational Mech Anal, 1989, 105: 243–266
[11] Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Lin´eaire, 1984, 1: 109–145
[12] Li, X. G. and Zhu, S. H. Blow-up rate for critical nonlinear Schr¨odinger equation with stark potential, Applicable Analysis, 2008, 87: 303–310
[13] Joly J L, Métivier G, Rauch J. Diffractive nonlinear geometric optics with rectification. Indiana Univ Math J, 1998, 47: 1167–1241
[14] Merle F. Tsutsumi Y. L2 concentration of blow up solutions for the nonlinear Schr¨odinger equation with critical power nonlinearity. J Differential Equations, 1990, 84: 205–214
[15] Merle F. On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schr¨odinger equation with critical exponent and critical mass. Comm Pure Appl Math, 1992, 45: 203–254
[16] Merle F. Determination of blow-up solutions with minimal mass for nonlinear Schr¨odinger equations with critical power. Duke Math J, 1993, 69: 427–454
[17] Merle F, Raphaël P. On universality of blow-up profile for L2 critical nonlinear Schr¨odinger equation. Invent Math, 2004, 156: 565–672
[18] Merle F, Raphaël P. Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr¨odinger equation. Annals of Math, 2005, 16: 157–222
[19] Nakamura Y. Local solvability and smoothing effects of nonlinear Schr¨odinger equations with magnetic fields. Funkcial Ekvac, 2001, 44: 1–18
[20] Nawa H. Asymptotic and limiting profiles of blowup solutions of the nonlinear Schr¨odinger equation with critical power. Comm Pure Appl Math, 1999, 52: 193–270
[21] Ogawa T, Tsutsumi Y. Blow-up of H1 solution for the nonlinear Schr¨odinger equation. J Differential Equations, 1991, 92: 317–330
[22] Ozawa T. Nonexistence of wave operators with Stark effect Hamiltonians. Math Z, 1991, 207: 335–339
[23] Ozawa T. Space-time behavior of propagators for Schr¨odinger evolution equations with Stark effect. J Funct Anal, 1991, 97: 264–292
[24] Sulem C, Sulem P L. The nonlinear Schr¨odinger equation, self-focusing and wave collapse. New York: Springer-Verlag, 1999
[25] Weinstein M I. Nonlinear Schr¨odinger equations and sharp interpolation estimates. Comm Math Phys, 1983, 87: 567–576
[26] Weinstein M I. On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Commun in PDE, 1986, 11: 545–565
[27] Tsurumi T, Wadati M. Free fall of atomic laser beam with weak inter-atomic interaction. J Phys Soc Japan, 2001, 70: 60–68
[28] Yajiam Y. Existence of solutions for Schr¨odinger evolution equations. Commun Math Phys, 1987, 110: 415–426
[29] Zhang J. Stability of attractive Bose-Einstein condensate. J Statist Phys, 2000, 101: 731–746
[30] Zhang J. Sharp conditions of global existence for nonlinear Schr¨odinger and Klein-Gordon equations. Nonlinear Anal, 2002, 48: 191–207
[31] Zhu S H, Zhang J. On the concentration properties for the nonlinear schrodinger equation with a start potential. Acta Mathematica Scientia, 2011, 31B(5): 1923–1938 |