# American Institute of Mathematical Sciences

April  2019, 39(4): 2187-2201. doi: 10.3934/dcds.2019092

## Existence of the normalized solutions to the nonlocal elliptic system with partial confinement

 Institute of Applied System Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, China

* Corresponding author: Jun Wang

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: This work was supported by NSFC grants: 11571140, 11671077, 11601190, Fellowship of Outstanding Young Scholars of Jiangsu Province(BK20160063), the Six big talent peaks project in Jiangsu Province(XYDXX-015), and NSF of Jiangsu Province Grants: BK20150478, BK20160483, and Jiangsu University Foundation Grant 16JDG043.

In present paper we study the existence and orbital stability of the standing waves to the nonlocal elliptic system with partial confinement. This type equations arises from the basic quantum chemistry model of small number of electrons interacting with static nucleii. On the one hand, we prove the existence of global minimizer of the associate energy functional subject to the $L^2$-constraint. On the other hand, we discuss the orbital stability of the global minimizers. Comparing to the local case, we need establish the new inequality related to the Steiner rearrangement.

Citation: Jun Wang, Qiuping Geng, Maochun Zhu. Existence of the normalized solutions to the nonlocal elliptic system with partial confinement. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2187-2201. doi: 10.3934/dcds.2019092
##### References:
 [1] J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Adv. Differential Equations, 18 (2013), 1129-1164.   Google Scholar [2] T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494.  Google Scholar [3] J. Bellazzini, N. Boussa, L. Jeanjean and N. Visciglia, Existence and Stability of Standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.  Google Scholar [4] 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 [5] V. Benci and N. Visciglia, Solitary waves with non-vanishing angular momentum, Adv. Nonlinear Stud., 3 (2013), 151-160.  doi: 10.1515/ans-2003-0104.  Google Scholar [6] S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities, Adv. Nonlinear Anal., 4 (2015), 73-90.  doi: 10.1515/anona-2014-0058.  Google Scholar [7] S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.  Google Scholar [8] H. Brezis and E. Lieb, 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 [9] J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differential Equations, 163 (2000), 429-474.  doi: 10.1006/jdeq.1999.3737.  Google Scholar [10] E. Cancés and C. Le Bris, On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics, Math. Models Methods Appl. Sci., 9 (1999), 963-990.  doi: 10.1142/S0218202599000440.  Google Scholar [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, J. Differential Equations, 246 (2009), 1921-1943.  doi: 10.1007/BF01403504.  Google Scholar [12] J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Mathematical Phys., 16 (1975), 1122-1130.  doi: 10.1063/1.522642.  Google Scholar [13] M. Colin, L. Jeanjean and M. Squassina, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, Nonlinearity, 23 (2010), 1353-1385.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar [14] B.-D. Esry, C.-H. Greene, J.-P. Burke Jr and J.-L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.  Google Scholar [15] D. Garrisi, On the orbital stability of standing-wave solutions to a coupled nonlinear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar [16] V. Georgiev and G. Venkov, Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential, J. Differential Equations, 251 (2011), 420-438.  doi: 10.1016/j.jde.2011.04.012.  Google Scholar [17] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.  Google Scholar [18] T.-X. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508, 12 pp. doi: 10.1063/1.5028208.  Google Scholar [19] T. Gou and L. 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Lions, Some remarks on Hartree equation, Nonlinear Anal., 5 (1981), 1245-1256.  doi: 10.1016/0362-546X(81)90016-X.  Google Scholar [25] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar [26] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar [27] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar [28] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [29] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar [30] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar [31] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp. doi: 10.1142/S0219199715500054.  Google Scholar [32] N.-V. Nguyen and Z.-Q. Wang, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Adv. Differential Equations, 16 (2011), 977-1000.   Google Scholar [33] N.-V. Nguyen and Z.-Q. Wang, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Nonlinear Anal., 90 (2013), 1-26.   Google Scholar [34] N.-V. Nguyen and Z.-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36 (2016), 1005-1021.  doi: 10.3934/dcds.2016.36.1005.  Google Scholar [35] M. Ohta, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Nonlinear Anal., 26 (1996), 933-939.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar [36] M. Shibata, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Manuscripta Math., 143 (2014), 221-237.  doi: 10.3934/dcds.2016.36.1005.  Google Scholar [37] M. Shibata, A new rearrangement inequality and its application for $L^2$-constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1.  Google Scholar [38] J. Wang, Standing waves solutions for the coupled Hartree-Fock type nonlocal elliptic system, Submitted, 2017. Google Scholar [39] J. Wang and Y.-Y. Dong, Bonded states solutions for a coupled nonlinear Hartree equations with nonlocal interaction, Submitted, 2017. Google Scholar [40] J. Wang and Q.-P. Geng, Existence and stability of standing waves for the Hartree equation with partial confinement, Submitted, 2018. Google Scholar [41] J. Wang and J.-P. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, 56 (2017), Art. 168, 36 pp. doi: 10.1007/s00526-017-1268-8.  Google Scholar [42] J. Wang and Z.-Q. Wang, Existence of odd solutions for the weakly coupled hartree type elliptic system with nonlocal interaction, Submitted, 2017. Google Scholar [43] J. Wang and W. Yang, Normalized solutions and asymptotical behavior of minimizer for the coupled Hartree equations, J. Differential Equations, 256 (2018), 501-544.  doi: 10.1016/j.jde.2018.03.003.  Google Scholar [44] J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys, 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar

show all references

##### References:
 [1] J. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, Adv. Differential Equations, 18 (2013), 1129-1164.   Google Scholar [2] T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494.  Google Scholar [3] J. Bellazzini, N. Boussa, L. Jeanjean and N. Visciglia, Existence and Stability of Standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.  Google Scholar [4] 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 [5] V. Benci and N. Visciglia, Solitary waves with non-vanishing angular momentum, Adv. Nonlinear Stud., 3 (2013), 151-160.  doi: 10.1515/ans-2003-0104.  Google Scholar [6] S. Bhattarai, Stability of solitary-wave solutions of coupled NLS equations with power-type nonlinearities, Adv. Nonlinear Anal., 4 (2015), 73-90.  doi: 10.1515/anona-2014-0058.  Google Scholar [7] S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.  Google Scholar [8] H. Brezis and E. Lieb, 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 [9] J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differential Equations, 163 (2000), 429-474.  doi: 10.1006/jdeq.1999.3737.  Google Scholar [10] E. Cancés and C. Le Bris, On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics, Math. Models Methods Appl. Sci., 9 (1999), 963-990.  doi: 10.1142/S0218202599000440.  Google Scholar [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, J. Differential Equations, 246 (2009), 1921-1943.  doi: 10.1007/BF01403504.  Google Scholar [12] J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Mathematical Phys., 16 (1975), 1122-1130.  doi: 10.1063/1.522642.  Google Scholar [13] M. Colin, L. Jeanjean and M. Squassina, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, Nonlinearity, 23 (2010), 1353-1385.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar [14] B.-D. Esry, C.-H. Greene, J.-P. Burke Jr and J.-L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.  Google Scholar [15] D. Garrisi, On the orbital stability of standing-wave solutions to a coupled nonlinear Klein-Gordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar [16] V. Georgiev and G. Venkov, Symmetry and uniqueness of minimizers of Hartree type equations with external Coulomb potential, J. Differential Equations, 251 (2011), 420-438.  doi: 10.1016/j.jde.2011.04.012.  Google Scholar [17] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.  Google Scholar [18] T.-X. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508, 12 pp. doi: 10.1063/1.5028208.  Google Scholar [19] T. Gou and L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144 (2016), 10-22.  doi: 10.1016/j.na.2016.05.016.  Google Scholar [20] N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14 (2014), 115-1362.  doi: 10.1515/ans-2014-0104.  Google Scholar [21] C. Le Bris and P.-L. Lions, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Bull. Amer. Math. Soc.(N.S.), 42 (2005), 291-363.   Google Scholar [22] E.-H. Lieb and B. Simon, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Comm. Math. Phys., 53 (1997), 185-194.   Google Scholar [23] E. H. Lieb and M. Loss, Analysis. Second edition. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [24] P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal., 5 (1981), 1245-1256.  doi: 10.1016/0362-546X(81)90016-X.  Google Scholar [25] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar [26] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar [27] P.-L. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case I, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar [28] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [29] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar [30] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.  Google Scholar [31] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp. doi: 10.1142/S0219199715500054.  Google Scholar [32] N.-V. Nguyen and Z.-Q. Wang, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Adv. Differential Equations, 16 (2011), 977-1000.   Google Scholar [33] N.-V. Nguyen and Z.-Q. Wang, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Nonlinear Anal., 90 (2013), 1-26.   Google Scholar [34] N.-V. Nguyen and Z.-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36 (2016), 1005-1021.  doi: 10.3934/dcds.2016.36.1005.  Google Scholar [35] M. Ohta, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Nonlinear Anal., 26 (1996), 933-939.  doi: 10.1016/0362-546X(94)00340-8.  Google Scholar [36] M. Shibata, Existence and stability of a two-parameter family of solitary waves for a 2-couple nonlinear Schrödinger system, Manuscripta Math., 143 (2014), 221-237.  doi: 10.3934/dcds.2016.36.1005.  Google Scholar [37] M. Shibata, A new rearrangement inequality and its application for $L^2$-constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1.  Google Scholar [38] J. Wang, Standing waves solutions for the coupled Hartree-Fock type nonlocal elliptic system, Submitted, 2017. Google Scholar [39] J. Wang and Y.-Y. Dong, Bonded states solutions for a coupled nonlinear Hartree equations with nonlocal interaction, Submitted, 2017. Google Scholar [40] J. Wang and Q.-P. Geng, Existence and stability of standing waves for the Hartree equation with partial confinement, Submitted, 2018. Google Scholar [41] J. Wang and J.-P. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, 56 (2017), Art. 168, 36 pp. doi: 10.1007/s00526-017-1268-8.  Google Scholar [42] J. Wang and Z.-Q. Wang, Existence of odd solutions for the weakly coupled hartree type elliptic system with nonlocal interaction, Submitted, 2017. Google Scholar [43] J. Wang and W. Yang, Normalized solutions and asymptotical behavior of minimizer for the coupled Hartree equations, J. Differential Equations, 256 (2018), 501-544.  doi: 10.1016/j.jde.2018.03.003.  Google Scholar [44] J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys, 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar
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