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Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations
1. | School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, China |
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.
doi: 10.1007/s00209-004-0663-y. |
[2] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271.
doi: 10.1007/s002050100152. |
[3] |
N. Ba, Y. Deng and S. Peng, Multi-peak bound states for Schröinger equations with compactly supported or unbounded potentials, Ann. I. H. Poincaré-AN, 27 (2010), 1205-1226.
doi: 10.1016/j.anihpc.2010.05.003. |
[4] |
S. Barile, A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008), 3525-3540.
doi: 10.1016/j.na.2007.03.044. |
[5] |
T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[6] |
D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424.
doi: 10.1016/j.jde.2005.06.027. |
[7] |
S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. Differential Equations, 188 (2003), 52-79.
doi: 10.1016/S0022-0396(02)00058-X. |
[8] |
S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[9] |
S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908. |
[10] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[11] |
S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108-130.
doi: 10.1016/S0022-247X(02)00278-0. |
[12] |
S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[13] |
M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.
doi: 10.1016/j.jmaa.2013.04.081. |
[14] |
Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Ann. Mat. Pura Appl., 190 (2011), 427-451.
doi: 10.1007/s10231-010-0157-y. |
[15] |
G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 251 (2011), 3500-3521.
doi: 10.1016/j.jde.2011.08.038. |
[16] |
E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105. |
[17] |
E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Mathematics 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[18] |
E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. |
[19] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[21] |
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. |
[22] |
V. Moroz and J. Van Schaftingen, Ground states 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. |
[23] |
M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal., 9 (2010), 1411-1419.
doi: 10.3934/cpaa.2010.9.1411. |
[24] |
R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[25] |
S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856.
doi: 10.1016/j.na.2010.01.021. |
[26] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905.
doi: 10.1063/1.3060169. |
[27] |
M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 403 (2013), 680-694.
doi: 10.1016/j.jmaa.2013.02.062. |
show all references
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.
doi: 10.1007/s00209-004-0663-y. |
[2] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271.
doi: 10.1007/s002050100152. |
[3] |
N. Ba, Y. Deng and S. Peng, Multi-peak bound states for Schröinger equations with compactly supported or unbounded potentials, Ann. I. H. Poincaré-AN, 27 (2010), 1205-1226.
doi: 10.1016/j.anihpc.2010.05.003. |
[4] |
S. Barile, A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008), 3525-3540.
doi: 10.1016/j.na.2007.03.044. |
[5] |
T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[6] |
D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424.
doi: 10.1016/j.jde.2005.06.027. |
[7] |
S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. Differential Equations, 188 (2003), 52-79.
doi: 10.1016/S0022-0396(02)00058-X. |
[8] |
S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[9] |
S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908. |
[10] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[11] |
S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108-130.
doi: 10.1016/S0022-247X(02)00278-0. |
[12] |
S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[13] |
M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.
doi: 10.1016/j.jmaa.2013.04.081. |
[14] |
Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Ann. Mat. Pura Appl., 190 (2011), 427-451.
doi: 10.1007/s10231-010-0157-y. |
[15] |
G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 251 (2011), 3500-3521.
doi: 10.1016/j.jde.2011.08.038. |
[16] |
E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105. |
[17] |
E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Mathematics 14, AMS, 2001.
doi: 10.1090/gsm/014. |
[18] |
E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. |
[19] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. |
[21] |
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. |
[22] |
V. Moroz and J. Van Schaftingen, Ground states 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. |
[23] |
M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal., 9 (2010), 1411-1419.
doi: 10.3934/cpaa.2010.9.1411. |
[24] |
R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[25] |
S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856.
doi: 10.1016/j.na.2010.01.021. |
[26] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905.
doi: 10.1063/1.3060169. |
[27] |
M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 403 (2013), 680-694.
doi: 10.1016/j.jmaa.2013.02.062. |
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