American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Resource allocation flowshop scheduling with learning effect and slack due window assignment
  • JIMO Home
  • This Issue
  • Next Article
    Two-stage mean-risk stochastic mixed integer optimization model for location-allocation problems under uncertain environment
September  2021, 17(5): 2805-2816. doi: 10.3934/jimo.2020095

Stability of ground state for the Schrödinger-Poisson equation

Qian Shen 1, and  Na Wei 2,,

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

* Corresponding author: Na Wei

Received  August 2019 Revised  March 2020 Published  September 2021 Early access  May 2020

Fund Project: The first author is supported by NSFC grant (No.11771314), the second author is supported by the Natural Science Foundation of Hubei Province (No.2019CFB570) and the Fundamental Research Funds for the Central Universities (No.2722019PY053)

We are concerned with the stability of the ground state for the Schrödinger-Poisson equation
$ 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. $
If
$ 2<p<\frac{7}{3} $
 and the frequency is sufficiently large, we show that the ground state is orbitally stable.
Keywords: Stability, ground state, Schrödinger-Poisson equation.
Mathematics Subject Classification: Primary: 93D99, 35Q41, 35Q55; Secondary: 35B35.
Citation: Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, Ph.D thesis, Tohoku University, Sendai, Japan, 2003.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

N. J. Mauser, The Schrödinger-Poisson-X equation, Appl. Math. Lett., 14 (2001), 759-763.  doi: 10.1016/S0893-9659(01)80038-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

R. Fukuizumi, Stability and instability of standing waves for nonlinear Schrödinger equations, Ph.D thesis, Tohoku University, Sendai, Japan, 2003.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

N. J. Mauser, The Schrödinger-Poisson-X equation, Appl. Math. Lett., 14 (2001), 759-763.  doi: 10.1016/S0893-9659(01)80038-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[2]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[3]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[4]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[5]

Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
[6]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure &amp; Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[7]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[8]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure &amp; Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[9]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184
[10]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure &amp; Applied Analysis, , () : -. doi: 10.3934/cpaa.2021108
[11]

Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure &amp; Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[12]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure &amp; Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867
[13]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
[14]

Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021
[15]

Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[16]

Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[17]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[18]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure &amp; Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039
[19]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038
[20]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure &amp; Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

2020 Impact Factor: 1.801

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences