American Institute of Mathematical Sciences

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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

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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

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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

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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

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