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Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity
1. | School of Mathematics and Computer Science, Fujian Normal University, Qishan Campus, Fuzhou 350117, China |
References:
[1] |
P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817.
|
[2] |
Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 879.
doi: 10.1088/0951-7715/21/5/001. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005).
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549.
doi: 10.1007/BF01403504. |
[5] |
G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential,, J. Math. Anal. Appl., 320 (2006), 591.
doi: 10.1016/j.jmaa.2005.07.008. |
[6] |
J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142.
doi: 10.1016/j.physd.2007.01.004. |
[7] |
J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential,, Commun. Math. Sci., 7 (2009), 549.
doi: 10.4310/CMS.2009.v7.n3.a2. |
[8] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33.
doi: 10.1016/0022-1236(79)90077-6. |
[9] |
R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794.
doi: 10.1063/1.523491. |
[10] |
E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231. Google Scholar |
[11] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2,, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109.
|
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003). Google Scholar |
[13] |
M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons,, SIAM J. Math. Anal., 36 (2004), 967.
doi: 10.1137/S0036141003431530. |
[14] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567.
|
[15] |
M. Willem, Minimax Theorems,, Birkhäuser, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731.
doi: 10.1023/A:1026437923987. |
show all references
References:
[1] |
P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation,, Adv. Math. Sci. Appl., 12 (2002), 817.
|
[2] |
Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 879.
doi: 10.1088/0951-7715/21/5/001. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Institute of Mathematical Sciences, 10 (2005).
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549.
doi: 10.1007/BF01403504. |
[5] |
G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential,, J. Math. Anal. Appl., 320 (2006), 591.
doi: 10.1016/j.jmaa.2005.07.008. |
[6] |
J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation,, Phys. D, 227 (2007), 142.
doi: 10.1016/j.physd.2007.01.004. |
[7] |
J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential,, Commun. Math. Sci., 7 (2009), 549.
doi: 10.4310/CMS.2009.v7.n3.a2. |
[8] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem,, J. Funct. Anal., 32 (1979), 33.
doi: 10.1016/0022-1236(79)90077-6. |
[9] |
R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation,, J. Math. Phys., 18 (1977), 1794.
doi: 10.1063/1.523491. |
[10] |
E. P. Gross, Physics of many-particle systems,, (eds. E. Meeron), 1 (1966), 231. Google Scholar |
[11] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2,, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109.
|
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics,, Vols. II, (2003). Google Scholar |
[13] |
M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons,, SIAM J. Math. Anal., 36 (2004), 967.
doi: 10.1137/S0036141003431530. |
[14] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (1983), 567.
|
[15] |
M. Willem, Minimax Theorems,, Birkhäuser, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Zhang, Stability of attractive Bose-Einstein condensates,, J. Statistical Phys., 101 (2000), 731.
doi: 10.1023/A:1026437923987. |
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