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January  2015, 35(1): 427-440. doi: 10.3934/dcds.2015.35.427

## 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

Received  November 2013 Revised  March 2014 Published  August 2014

In this paper, we obtain the existence of infinitely many solutions for the following Schrödinger-Poisson system \begin{equation*} \begin{cases} -\Delta u+a(x)u+ \phi u=k(x)|u|^{q-2}u- h(x)|u|^{p-2}u,\quad &x\in \mathbb{R}^3,\\ -\Delta \phi=u^2,\ \lim_{|x|\to +\infty}\phi(x)=0, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $1 < q < 2 < p < +\infty$, $a(x)$, $k(x)$ and $h(x)$ are measurable functions satisfying suitable assumptions.
Citation: Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
##### 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 $\mathbb{R}^3$, 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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##### 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 $\mathbb{R}^3$, 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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