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]

C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 437.  doi: 10.1007/s00030-005-0021-8.  Google Scholar

[2]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method,, Comm. Pure. Appl. Anal., 8 (2009), 1745.  doi: 10.3934/cpaa.2009.8.1745.  Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $R^N$,, J. Differential Equations, 246 (2009), 1288.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch. Ration. Mech. Anal., 159 (2001), 253.  doi: 10.1007/s002050100152.  Google Scholar

[5]

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

[6]

A. Azzollini and A. Pomponio, A note on the ground state solutions for the nonlinear Schrödinger-Maxwell equations,, Boll. Unione Mat. Ital., 9 (2009), 93.   Google Scholar

[7]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology,, Cal. Var. Partial Differential Equations, 2 (1994), 29.  doi: 10.1007/BF01234314.  Google Scholar

[8]

R. Benguria and H. Brezis, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules,, Comm. Math. Phys., 79 (1981), 167.   Google Scholar

[9]

J. Byeon and Z. Q. Wang, Standing waves for nonlinear Schrödinger equations with singular potentials,, Ann. I. H. Poincar\'e-AN, 26 (2009), 943.  doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

[10]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1.   Google Scholar

[11]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, Commun. Appl. Anal., 7 (2003), 417.   Google Scholar

[12]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system,, Comm. Partial Differential Equations, 17 (1992), 1051.  doi: 10.1080/03605309208820878.  Google Scholar

[13]

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

[14]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Advanced Nonlinear Studies, 8 (2008), 573.   Google Scholar

[15]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, I. Necessary conditions,, Math. Models Methods Appl. Sci., 19 (2009), 707.  doi: 10.1142/S0218202509003589.  Google Scholar

[16]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, II. Existence,, Math. Models Methods Appl. Sci., 19 (2009), 877.  doi: 10.1142/S0218202509003656.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. II,, Ann. Inst. H. Poincare Anal. Non Lin\'eaire, 1 (1984), 223.   Google Scholar

[18]

M. Del Pino and P. Felmer, Semiclassical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.   Google Scholar

[19]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, Journ. Func. Anal., 237 (2006), 655.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141.   Google Scholar

[21]

O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system,, J. Statist. Phys., 114 (2004), 179.  doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[22]

M. Yang, Z. Shen and Y. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system,, Nonlinear Analysis, 71 (2009), 730.  doi: 10.1016/j.na.2008.10.105.  Google Scholar

[23]

H. Yin and P. Chang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity,, J. Differential Equations, 247 (2009), 618.  doi: 10.1016/j.jde.2009.03.002.  Google Scholar

[24]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent,, Nonlinear Analysis, 70 (2009), 2150.  doi: 10.1016/j.na.2008.02.116.  Google Scholar

show all references

References:
[1]

C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 437.  doi: 10.1007/s00030-005-0021-8.  Google Scholar

[2]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method,, Comm. Pure. Appl. Anal., 8 (2009), 1745.  doi: 10.3934/cpaa.2009.8.1745.  Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $R^N$,, J. Differential Equations, 246 (2009), 1288.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, Arch. Ration. Mech. Anal., 159 (2001), 253.  doi: 10.1007/s002050100152.  Google Scholar

[5]

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

[6]

A. Azzollini and A. Pomponio, A note on the ground state solutions for the nonlinear Schrödinger-Maxwell equations,, Boll. Unione Mat. Ital., 9 (2009), 93.   Google Scholar

[7]

V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology,, Cal. Var. Partial Differential Equations, 2 (1994), 29.  doi: 10.1007/BF01234314.  Google Scholar

[8]

R. Benguria and H. Brezis, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules,, Comm. Math. Phys., 79 (1981), 167.   Google Scholar

[9]

J. Byeon and Z. Q. Wang, Standing waves for nonlinear Schrödinger equations with singular potentials,, Ann. I. H. Poincar\'e-AN, 26 (2009), 943.  doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

[10]

S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1.   Google Scholar

[11]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, Commun. Appl. Anal., 7 (2003), 417.   Google Scholar

[12]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system,, Comm. Partial Differential Equations, 17 (1992), 1051.  doi: 10.1080/03605309208820878.  Google Scholar

[13]

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

[14]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Advanced Nonlinear Studies, 8 (2008), 573.   Google Scholar

[15]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, I. Necessary conditions,, Math. Models Methods Appl. Sci., 19 (2009), 707.  doi: 10.1142/S0218202509003589.  Google Scholar

[16]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, II. Existence,, Math. Models Methods Appl. Sci., 19 (2009), 877.  doi: 10.1142/S0218202509003656.  Google Scholar

[17]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. II,, Ann. Inst. H. Poincare Anal. Non Lin\'eaire, 1 (1984), 223.   Google Scholar

[18]

M. Del Pino and P. Felmer, Semiclassical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.   Google Scholar

[19]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, Journ. Func. Anal., 237 (2006), 655.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[20]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141.   Google Scholar

[21]

O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system,, J. Statist. Phys., 114 (2004), 179.  doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[22]

M. Yang, Z. Shen and Y. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system,, Nonlinear Analysis, 71 (2009), 730.  doi: 10.1016/j.na.2008.10.105.  Google Scholar

[23]

H. Yin and P. Chang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity,, J. Differential Equations, 247 (2009), 618.  doi: 10.1016/j.jde.2009.03.002.  Google Scholar

[24]

L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent,, Nonlinear Analysis, 70 (2009), 2150.  doi: 10.1016/j.na.2008.02.116.  Google Scholar

[1]

Pingzheng Zhang, Jianhua Sun. Clustered layers for the Schrödinger-Maxwell system on a ball. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 657-688. doi: 10.3934/dcds.2006.16.657
[2]

Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117
[3]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055
[4]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[5]

Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040
[6]

Tai-Chia Lin, Tsung-Fang Wu. Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110
[7]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
[8]

Gan Lu, Weiming Liu. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096
[9]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[10]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[12]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[13]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[14]

Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
[15]

Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411
[16]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[17]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022
[18]

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
[19]

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
[20]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences