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November  2016, 36(11): 5881-5910. doi: 10.3934/dcds.2016058

## Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$

 1 Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande - Pb, Brazil 2 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Received  June 2015 Revised  April 2016 Published  August 2016

In this paper we are going to study a class of Schrödinger-Poisson system $$\left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=4\pi u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right.$$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint bounded components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.
Citation: Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
##### References:
 [1] C. O. Alves, Existence of multi-bump solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 6 (2006), 491-509. [2] C. O. Alves, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems in $\mathbb{R}^N2$, Top. Meth. Nonlinear Anal., 34 (2009), 231-250. doi: 10.12775/TMNA.2009.040. [3] C. O. Alves and M. A. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. [4] A. Ambrosetti and R. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. [5] 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. [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonlinear Anal., 11 (1998), 283-293. [7] H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I - existence of a ground state, Arch. Rat. Mech. Analysis, 82 (1983), 313-346. doi: 10.1007/BF00250555. [8] O. Bokanowski and N. J. Mauser, Local approximation for the Hartree-Fock exchange potential: A deformation approach, $M^3$AS, 9 (1999), 941-961. doi: 10.1142/S0218202599000439. [9] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [10] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. [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] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. [13] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [14] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDE, 4 (1996), 121-137. doi: 10.1007/BF01189950. [15] Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. [16] L. Gongbao, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math., 14 (1989), 27-36. doi: 10.5186/aasfm.1990.1521. [17] G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Analysis and Appl., 365 (2010), 288-299. doi: 10.1016/j.jmaa.2009.10.061. [18] H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. doi: 10.1016/j.na.2006.07.029. [19] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. [20] I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. [21] Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [22] S. Kim and J. Seok, On nodal solutions of the Nonlinear Schrödinger-Poisson equations, Comm. Cont. Math., 14 (2012), 12450041-12450057. doi: 10.1142/S0219199712500411. [23] C. Miranda, Un' osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7. [24] N. J. Mauser, The Schrödinger-Poisson-$X_\alpha$ equation, Applied Math. Letters, 14 (2001), 759-763. doi: 10.1016/S0893-9659(01)80038-0. [25] L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528. doi: 10.1016/j.aml.2007.06.005. [26] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [27] D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. [28] O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statistical Physics, 114 (2004), 179-204. doi: 10.1023/B:JOSS.0000003109.97208.53. [29] E. Séré, Existence of infinitely many homoclinic orbits in Halmitonian systems, Math. Z., 209 (1992), 27-42. doi: 10.1007/BF02570817. [30] F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116. [31] X. Zhang and S. Ma, Multi-bump solutions of Schrödinger-Poisson equations with steep potential well, Z. Angew. Math. Phys., 66 (2015), 1615-1631. doi: 10.1007/s00033-014-0490-x. [32] M. Willem, Minimax Theorems, Birkhäuser Boston, MA 1996. doi: 10.1007/978-1-4612-4146-1.

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##### References:
 [1] C. O. Alves, Existence of multi-bump solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 6 (2006), 491-509. [2] C. O. Alves, Multiplicity of multi-bump type nodal solutions for a class of elliptic problems in $\mathbb{R}^N2$, Top. Meth. Nonlinear Anal., 34 (2009), 231-250. doi: 10.12775/TMNA.2009.040. [3] C. O. Alves and M. A. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. [4] A. Ambrosetti and R. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. [5] 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. [6] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonlinear Anal., 11 (1998), 283-293. [7] H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I - existence of a ground state, Arch. Rat. Mech. Analysis, 82 (1983), 313-346. doi: 10.1007/BF00250555. [8] O. Bokanowski and N. J. Mauser, Local approximation for the Hartree-Fock exchange potential: A deformation approach, $M^3$AS, 9 (1999), 941-961. doi: 10.1142/S0218202599000439. [9] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017. [10] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. [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] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. [13] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [14] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDE, 4 (1996), 121-137. doi: 10.1007/BF01189950. [15] Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. [16] L. Gongbao, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math., 14 (1989), 27-36. doi: 10.5186/aasfm.1990.1521. [17] G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Analysis and Appl., 365 (2010), 288-299. doi: 10.1016/j.jmaa.2009.10.061. [18] H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. doi: 10.1016/j.na.2006.07.029. [19] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. [20] I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. [21] Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [22] S. Kim and J. Seok, On nodal solutions of the Nonlinear Schrödinger-Poisson equations, Comm. Cont. Math., 14 (2012), 12450041-12450057. doi: 10.1142/S0219199712500411. [23] C. Miranda, Un' osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7. [24] N. J. Mauser, The Schrödinger-Poisson-$X_\alpha$ equation, Applied Math. Letters, 14 (2001), 759-763. doi: 10.1016/S0893-9659(01)80038-0. [25] L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528. doi: 10.1016/j.aml.2007.06.005. [26] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005. [27] D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. [28] O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statistical Physics, 114 (2004), 179-204. doi: 10.1023/B:JOSS.0000003109.97208.53. [29] E. Séré, Existence of infinitely many homoclinic orbits in Halmitonian systems, Math. Z., 209 (1992), 27-42. doi: 10.1007/BF02570817. [30] F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116. [31] X. Zhang and S. Ma, Multi-bump solutions of Schrödinger-Poisson equations with steep potential well, Z. Angew. Math. Phys., 66 (2015), 1615-1631. doi: 10.1007/s00033-014-0490-x. [32] M. Willem, Minimax Theorems, Birkhäuser Boston, MA 1996. doi: 10.1007/978-1-4612-4146-1.
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