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
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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

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