American Institute of Mathematical Sciences

Advanced
July  2011, 10(4): 1267-1279. doi: 10.3934/cpaa.2011.10.1267

Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system

Yanqin Fang 1, and  Jihui Zhang 1,

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, China, China

Received  April 2010 Revised  November 2010 Published  April 2011

In this paper, we study the following system

$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

Keywords: multiple solutions., Ljusternik-Schnirelmann theory, Schrödinger-Maxwell.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J10, 35B3.
Citation: Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
References:
[1]

Nonlinear Differ. Equ. Appl., 12 (2005), 437-457. doi: 10.1007/s00030-005-0021-8.  Google Scholar

[2]

Comm. Pure. Appl. Anal., 8 (2009), 1745-1758. doi: 10.3934/cpaa.2009.8.1745.  Google Scholar

[3]

J. Differential Equations, 246 (2009), 1288-1311. doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[5]

J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[6]

Boll. Unione Mat. Ital., 9 (2009), 93-104.  Google Scholar

[7]

Cal. Var. Partial Differential Equations, 2 (1994), 29-48. doi: 10.1007/BF01234314.  Google Scholar

[8]

Comm. Math. Phys., 79 (1981), 167-180.  Google Scholar

[9]

Ann. I. H. Poincaré-AN, 26 (2009), 943-958. doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

[10]

Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  Google Scholar

[11]

Commun. Appl. Anal., 7 (2003), 417-423.  Google Scholar

[12]

Comm. Partial Differential Equations, 17 (1992), 1051-1110. doi: 10.1080/03605309208820878.  Google Scholar

[13]

Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.  Google Scholar

[14]

Advanced Nonlinear Studies, 8 (2008), 573-595.  Google Scholar

[15]

Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589.  Google Scholar

[16]

Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656.  Google Scholar

[17]

Ann. Inst. H. Poincare Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[18]

J. Funct. Anal., 149 (1997), 245-265.  Google Scholar

[19]

Journ. Func. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

Math. Models Methods Appl. Sci., 15 (2005), 141-164.  Google Scholar

[21]

J. Statist. Phys., 114 (2004), 179-204. doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[22]

Nonlinear Analysis, 71 (2009), 730-739. doi: 10.1016/j.na.2008.10.105.  Google Scholar

[23]

J. Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.  Google Scholar

[24]

Nonlinear Analysis, 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.  Google Scholar

show all references

References:
[1]

Nonlinear Differ. Equ. Appl., 12 (2005), 437-457. doi: 10.1007/s00030-005-0021-8.  Google Scholar

[2]

Comm. Pure. Appl. Anal., 8 (2009), 1745-1758. doi: 10.3934/cpaa.2009.8.1745.  Google Scholar

[3]

J. Differential Equations, 246 (2009), 1288-1311. doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[5]

J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar

[6]

Boll. Unione Mat. Ital., 9 (2009), 93-104.  Google Scholar

[7]

Cal. Var. Partial Differential Equations, 2 (1994), 29-48. doi: 10.1007/BF01234314.  Google Scholar

[8]

Comm. Math. Phys., 79 (1981), 167-180.  Google Scholar

[9]

Ann. I. H. Poincaré-AN, 26 (2009), 943-958. doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

[10]

Topol. Methods Nonlinear Anal., 10 (1997), 1-13.  Google Scholar

[11]

Commun. Appl. Anal., 7 (2003), 417-423.  Google Scholar

[12]

Comm. Partial Differential Equations, 17 (1992), 1051-1110. doi: 10.1080/03605309208820878.  Google Scholar

[13]

Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.  Google Scholar

[14]

Advanced Nonlinear Studies, 8 (2008), 573-595.  Google Scholar

[15]

Math. Models Methods Appl. Sci., 19 (2009), 707-720. doi: 10.1142/S0218202509003589.  Google Scholar

[16]

Math. Models Methods Appl. Sci., 19 (2009), 877-910. doi: 10.1142/S0218202509003656.  Google Scholar

[17]

Ann. Inst. H. Poincare Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[18]

J. Funct. Anal., 149 (1997), 245-265.  Google Scholar

[19]

Journ. Func. Anal., 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

Math. Models Methods Appl. Sci., 15 (2005), 141-164.  Google Scholar

[21]

J. Statist. Phys., 114 (2004), 179-204. doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[22]

Nonlinear Analysis, 71 (2009), 730-739. doi: 10.1016/j.na.2008.10.105.  Google Scholar

[23]

J. Differential Equations, 247 (2009), 618-647. doi: 10.1016/j.jde.2009.03.002.  Google Scholar

[24]

Nonlinear Analysis, 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.  Google Scholar

[1]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[2]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[3]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047
[4]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054
[5]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115
[6]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018
[7]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011
[8]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061
[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[10]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024
[11]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450
[13]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[14]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028
[15]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[16]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039
[17]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038
[18]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[19]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
[20]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences