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Stability of ground state for the Schrödinger-Poisson equation
1. | School of Mathematical Sciencesand V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, 610066, China |
2. | School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China |
$ i\frac{\partial\psi}{\partial t}+\triangle\psi-(|x|^{-1}\ast|\psi|^2)\psi+|\psi|^{p-1}\psi = 0,\quad x\in \mathbb{R}^3. $ |
$ 2<p<\frac{7}{3} $ |
References:
[1] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[2] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[3] |
J. Bellazzini and G. Siciliano,
Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 267-280.
doi: 10.1007/s00033-010-0092-1. |
[4] |
J. Bellazzini, L. Jeanjean and T. Luo,
Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303-339.
doi: 10.1112/plms/pds072. |
[5] |
O. Bokanowski, J. L. López and J. Soler,
On an exchange interaction model for quantum transport: The Schrödinger-Poisson- Slater system, Math. Models Methods Appl. Sci., 13 (2003), 1397-1412.
doi: 10.1142/S0218202503002969. |
[6] |
O. Bokanowski and N. J. Mauser,
Local approximation for the Hartree-Fock exchange potential: A deformation approach, Math. Models Methods Appl. Sci., 9 (1999), 941-961.
doi: 10.1142/S0218202599000439. |
[7] |
H. Brézis and E. H. Leib,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[8] |
T. Cazenave, Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[9] |
T. Cazenave and P.-L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[10] |
R. Fukuizumi,
Remarks on the stable standing waves for nonlinear Schrödinger equation with double power nonlinearity, Adv. Math. Sci. Appl., 13 (2003), 549-564.
|
[11] |
R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, Ph.D thesis, Tohoku University, Sendai, Japan, 2003. |
[12] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry (Ⅰ), J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[13] |
L. Jeanjean and S. Le Coz,
An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.
|
[14] |
L. Jeanjean and T. Luo,
Sharp nonexistence results of prescried $L^2-$ norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.
doi: 10.1007/s00033-012-0272-2. |
[15] |
Y. S. Jiang, Z. P. Wang and H. S. Zhou,
Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Nonlinear Anal., 83 (2013), 50-57.
doi: 10.1016/j.na.2013.01.006. |
[16] |
Y. S. Jiang and H. S. Zhou,
Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential, Sci. China Math., 57 (2014), 1163-1174.
doi: 10.1007/s11425-014-4790-6. |
[17] |
Y. S. Jiang and H. S. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[18] |
H. Kikuchi,
Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
doi: 10.1515/ans-2007-0305. |
[19] |
M. K. Kwong,
Uniqueness of positive solutions of $\bigtriangleup u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[20] |
C. Lavor, L. Libeti, N. Maculan and M. A. C. Nascimento, Solving Hartree-Fock systems with global optimization methods, Europhys. Lett. EPL, 77 (2007), 50006, 5 pp.
doi: 10.1209/0295-5075/77/50006. |
[21] |
E. H. Leib,
Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[22] |
X. G. Li, J. Zhang and Y. H. Wu,
Strong instability of standing waves for the Schrödinger-Poisson-Slater equation (in Chinese), Sci. Sin. Math., 46 (2016), 45-58.
doi: 10.1360/N012014-00007. |
[23] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations: The locally compact case 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[24] |
O. Lopes and M. Maris,
Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.
doi: 10.1016/j.jfa.2007.10.004. |
[25] |
N. J. Mauser,
The Schrödinger-Poisson-X equation, Appl. Math. Lett., 14 (2001), 759-763.
doi: 10.1016/S0893-9659(01)80038-0. |
[26] |
J. F. Mennemann, D. Matthes, R. M. Weishäupl and T. Langen, Optimal control of Bose-Einstein condensates in three dimensions, New J. Phys., 17 (2015), 113027.
doi: 10.1088/1367-2630/17/11/113027. |
[27] |
S. Pötting, M. Cramer and P. Meystre, Momentum-state engineering and control in Bose-Einstein condensates, Phys. Rev. A, 64 (2001), 063613. Google Scholar |
[28] |
L. Rosier and B.-Y. Zhang,
Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153.
doi: 10.1016/j.jde.2008.11.004. |
[29] |
D. Ruiz,
Schrödinger-Poisson equation under the effect of nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[30] |
O. Sánchez and J. Soler,
Long-time dynamics of Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.
doi: 10.1023/B:JOSS.0000003109.97208.53. |
show all references
References:
[1] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[2] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[3] |
J. Bellazzini and G. Siciliano,
Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 267-280.
doi: 10.1007/s00033-010-0092-1. |
[4] |
J. Bellazzini, L. Jeanjean and T. Luo,
Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303-339.
doi: 10.1112/plms/pds072. |
[5] |
O. Bokanowski, J. L. López and J. Soler,
On an exchange interaction model for quantum transport: The Schrödinger-Poisson- Slater system, Math. Models Methods Appl. Sci., 13 (2003), 1397-1412.
doi: 10.1142/S0218202503002969. |
[6] |
O. Bokanowski and N. J. Mauser,
Local approximation for the Hartree-Fock exchange potential: A deformation approach, Math. Models Methods Appl. Sci., 9 (1999), 941-961.
doi: 10.1142/S0218202599000439. |
[7] |
H. Brézis and E. H. Leib,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[8] |
T. Cazenave, Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[9] |
T. Cazenave and P.-L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[10] |
R. Fukuizumi,
Remarks on the stable standing waves for nonlinear Schrödinger equation with double power nonlinearity, Adv. Math. Sci. Appl., 13 (2003), 549-564.
|
[11] |
R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, Ph.D thesis, Tohoku University, Sendai, Japan, 2003. |
[12] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry (Ⅰ), J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[13] |
L. Jeanjean and S. Le Coz,
An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.
|
[14] |
L. Jeanjean and T. Luo,
Sharp nonexistence results of prescried $L^2-$ norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.
doi: 10.1007/s00033-012-0272-2. |
[15] |
Y. S. Jiang, Z. P. Wang and H. S. Zhou,
Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^3$, Nonlinear Anal., 83 (2013), 50-57.
doi: 10.1016/j.na.2013.01.006. |
[16] |
Y. S. Jiang and H. S. Zhou,
Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential, Sci. China Math., 57 (2014), 1163-1174.
doi: 10.1007/s11425-014-4790-6. |
[17] |
Y. S. Jiang and H. S. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[18] |
H. Kikuchi,
Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
doi: 10.1515/ans-2007-0305. |
[19] |
M. K. Kwong,
Uniqueness of positive solutions of $\bigtriangleup u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[20] |
C. Lavor, L. Libeti, N. Maculan and M. A. C. Nascimento, Solving Hartree-Fock systems with global optimization methods, Europhys. Lett. EPL, 77 (2007), 50006, 5 pp.
doi: 10.1209/0295-5075/77/50006. |
[21] |
E. H. Leib,
Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[22] |
X. G. Li, J. Zhang and Y. H. Wu,
Strong instability of standing waves for the Schrödinger-Poisson-Slater equation (in Chinese), Sci. Sin. Math., 46 (2016), 45-58.
doi: 10.1360/N012014-00007. |
[23] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations: The locally compact case 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[24] |
O. Lopes and M. Maris,
Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.
doi: 10.1016/j.jfa.2007.10.004. |
[25] |
N. J. Mauser,
The Schrödinger-Poisson-X equation, Appl. Math. Lett., 14 (2001), 759-763.
doi: 10.1016/S0893-9659(01)80038-0. |
[26] |
J. F. Mennemann, D. Matthes, R. M. Weishäupl and T. Langen, Optimal control of Bose-Einstein condensates in three dimensions, New J. Phys., 17 (2015), 113027.
doi: 10.1088/1367-2630/17/11/113027. |
[27] |
S. Pötting, M. Cramer and P. Meystre, Momentum-state engineering and control in Bose-Einstein condensates, Phys. Rev. A, 64 (2001), 063613. Google Scholar |
[28] |
L. Rosier and B.-Y. Zhang,
Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153.
doi: 10.1016/j.jde.2008.11.004. |
[29] |
D. Ruiz,
Schrödinger-Poisson equation under the effect of nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[30] |
O. Sánchez and J. Soler,
Long-time dynamics of Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.
doi: 10.1023/B:JOSS.0000003109.97208.53. |
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