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Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities
1. | School of Mathematical Sciences, Capital Normal University, Beijing 100037, China |
2. | Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China |
References:
[1] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal. TMA, 74 (2011), 5705-5721.
doi: 10.1016/j.na.2011.05.057. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schroinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincare Anal. NonLineaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[4] |
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. |
[5] |
T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.
doi: 10.1142/S0219199701000494. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[7] |
K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002, 65-76. |
[8] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[9] |
S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations, Nonlinear Anal. TMA, 71 (2009), 4927-4934.
doi: 10.1016/j.na.2009.03.050. |
[10] |
L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[11] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[12] |
D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[13] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schröinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[14] |
H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal. TMA, 67 (2007), 1445-1456.
doi: 10.1016/j.na.2006.07.029. |
[15] |
A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms, Nonlinear Anal. Real World Applications, 13 (2012), 213-223.
doi: 10.1016/j.nonrwa.2011.07.027. |
[16] |
G. B. Li, S. J. Peng and S. S. Yan., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.
doi: 10.1142/S0219199710004068. |
[17] |
P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1984), 33-97.
doi: 10.1007/BF01205672. |
[18] |
S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities, Nonlinear Anal. TMA, 53 (2003), 723-731.
doi: 10.1016/S0362-546X(03)00020-8. |
[19] |
D. Ruiz, The 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. |
[20] |
J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.
doi: 10.1016/j.jmaa.2012.01.057. |
[21] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[22] |
E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities, Topol. Methods Nonlinear Anal., 13 (1999), 251-271. |
[23] |
Z. P. Wang and H. S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[24] |
Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differ. Equ. Appl., 8 (2001), 15-33.
doi: 10.1007/PL00001436. |
[25] |
M. B. Yang, Z. F. Shen and Y. H. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system, Nonlinear Anal., 71 (2009), 730-739.
doi: 10.1016/j.na.2008.10.105. |
[26] |
L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal. TMA, 74 (2011), 5705-5721.
doi: 10.1016/j.na.2011.05.057. |
[3] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schroinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincare Anal. NonLineaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[4] |
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. |
[5] |
T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.
doi: 10.1142/S0219199701000494. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[7] |
K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002, 65-76. |
[8] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[9] |
S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations, Nonlinear Anal. TMA, 71 (2009), 4927-4934.
doi: 10.1016/j.na.2009.03.050. |
[10] |
L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493-516.
doi: 10.1016/S0294-1449(98)80032-2. |
[11] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[12] |
D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[13] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schröinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[14] |
H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal. TMA, 67 (2007), 1445-1456.
doi: 10.1016/j.na.2006.07.029. |
[15] |
A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms, Nonlinear Anal. Real World Applications, 13 (2012), 213-223.
doi: 10.1016/j.nonrwa.2011.07.027. |
[16] |
G. B. Li, S. J. Peng and S. S. Yan., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.
doi: 10.1142/S0219199710004068. |
[17] |
P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1984), 33-97.
doi: 10.1007/BF01205672. |
[18] |
S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities, Nonlinear Anal. TMA, 53 (2003), 723-731.
doi: 10.1016/S0362-546X(03)00020-8. |
[19] |
D. Ruiz, The 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. |
[20] |
J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.
doi: 10.1016/j.jmaa.2012.01.057. |
[21] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[22] |
E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities, Topol. Methods Nonlinear Anal., 13 (1999), 251-271. |
[23] |
Z. P. Wang and H. S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[24] |
Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differ. Equ. Appl., 8 (2001), 15-33.
doi: 10.1007/PL00001436. |
[25] |
M. B. Yang, Z. F. Shen and Y. H. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system, Nonlinear Anal., 71 (2009), 730-739.
doi: 10.1016/j.na.2008.10.105. |
[26] |
L. G. Zhao and F. K. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
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