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

Claudianor O.  Alves; Minbo  Yang

doi:10.3934/dcds.2016058

Article Contents

2016, Volume 36, Issue 11: 5881-5910. Doi: 10.3934/dcds.2016058

This issue Previous Article Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes Next Article Discrete and continuous topological dynamics: Fields of cross sections and expansive flows

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

Claudianor O. Alves^1, and
Minbo Yang^2,

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

Received: June 2015

Revised: April 2016

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 35J20, 35J65.

Citation:

Related Papers

Cited by

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 $\mathbbR^N$, 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.