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September  2021, 17(5): 2805-2816. doi: 10.3934/jimo.2020095

## 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

* 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 and the frequency is sufficiently large, we show that the ground state is orbitally stable. Citation: Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial and 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. [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. [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. [28] L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation, J. 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