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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 & Continuous Dynamical Systems - A, 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. doi: 10.1142/S021919970800282X. Google Scholar [2] P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system,, Nonlinear Anal. TMA, 74 (2011), 5705. doi: 10.1016/j.na.2011.05.057. Google Scholar [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. doi: 10.1016/j.anihpc.2009.11.012. Google Scholar [4] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057. Google Scholar [5] T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283. Google Scholar [7] K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$,, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, (2002), 65. Google Scholar [8] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [9] S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations,, Nonlinear Anal. TMA, 71 (2009), 4927. doi: 10.1016/j.na.2009.03.050. Google Scholar [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. doi: 10.1016/S0294-1449(98)80032-2. Google Scholar [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. doi: 10.1017/S030821050000353X. Google Scholar [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. doi: 10.1016/S0022-1236(02)00060-5. Google Scholar [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). doi: 10.1063/1.3683156. Google Scholar [14] H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal. TMA, 67 (2007), 1445. doi: 10.1016/j.na.2006.07.029. Google Scholar [15] A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms,, Nonlinear Anal. Real World Applications, 13 (2012), 213. doi: 10.1016/j.nonrwa.2011.07.027. Google Scholar [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. doi: 10.1142/S0219199710004068. Google Scholar [17] P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1984), 33. doi: 10.1007/BF01205672. Google Scholar [18] S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities,, Nonlinear Anal. TMA, 53 (2003), 723. doi: 10.1016/S0362-546X(03)00020-8. Google Scholar [19] D. Ruiz, The Schrödinger-Poisson equation under the effect of nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar [20] J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 390 (2012), 514. doi: 10.1016/j.jmaa.2012.01.057. Google Scholar [21] W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [22] E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities,, Topol. Methods Nonlinear Anal., 13 (1999), 251. Google Scholar [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. doi: 10.3934/dcds.2007.18.809. Google Scholar [24] Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, Nonlinear Differ. Equ. Appl., 8 (2001), 15. doi: 10.1007/PL00001436. Google Scholar [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. doi: 10.1016/j.na.2008.10.105. Google Scholar [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. doi: 10.1016/j.jmaa.2008.04.053. Google Scholar

show all references

##### References:
 [1] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar [2] P. d'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system,, Nonlinear Anal. TMA, 74 (2011), 5705. doi: 10.1016/j.na.2011.05.057. Google Scholar [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. doi: 10.1016/j.anihpc.2009.11.012. Google Scholar [4] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057. Google Scholar [5] T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283. Google Scholar [7] K. Benmlih, Stationary solutions for a Schröinger-Poisson system in $\mathbbR^3$,, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, (2002), 65. Google Scholar [8] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [9] S. J. Chen and C. L. Tang, High energy solutions for the superlinear Schrödinger Maxwell equations,, Nonlinear Anal. TMA, 71 (2009), 4927. doi: 10.1016/j.na.2009.03.050. Google Scholar [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. doi: 10.1016/S0294-1449(98)80032-2. Google Scholar [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. doi: 10.1017/S030821050000353X. Google Scholar [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. doi: 10.1016/S0022-1236(02)00060-5. Google Scholar [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). doi: 10.1063/1.3683156. Google Scholar [14] H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal. TMA, 67 (2007), 1445. doi: 10.1016/j.na.2006.07.029. Google Scholar [15] A. Kristály and D. Repovš, On the Schröinger-Maxwell system involving sublinear terms,, Nonlinear Anal. Real World Applications, 13 (2012), 213. doi: 10.1016/j.nonrwa.2011.07.027. Google Scholar [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. doi: 10.1142/S0219199710004068. Google Scholar [17] P. L. Lions, Solutios of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1984), 33. doi: 10.1007/BF01205672. Google Scholar [18] S. B. Liu and S. J. Li, An elliptic equation with concave and convex nonlinearities,, Nonlinear Anal. TMA, 53 (2003), 723. doi: 10.1016/S0362-546X(03)00020-8. Google Scholar [19] D. Ruiz, The Schrödinger-Poisson equation under the effect of nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar [20] J. T. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 390 (2012), 514. doi: 10.1016/j.jmaa.2012.01.057. Google Scholar [21] W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [22] E. Tonkes, A semilinear elliptic equation with concave and convex nonlinearities,, Topol. Methods Nonlinear Anal., 13 (1999), 251. Google Scholar [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. doi: 10.3934/dcds.2007.18.809. Google Scholar [24] Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, Nonlinear Differ. Equ. Appl., 8 (2001), 15. doi: 10.1007/PL00001436. Google Scholar [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. doi: 10.1016/j.na.2008.10.105. Google Scholar [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. doi: 10.1016/j.jmaa.2008.04.053. Google Scholar
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