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Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system
1. | Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, China, China |
$-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$
$-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$
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.
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show all references
References:
[1] |
Nonlinear Differ. Equ. Appl., 12 (2005), 437-457.
doi: 10.1007/s00030-005-0021-8. |
[2] |
Comm. Pure. Appl. Anal., 8 (2009), 1745-1758.
doi: 10.3934/cpaa.2009.8.1745. |
[3] |
J. Differential Equations, 246 (2009), 1288-1311.
doi: 10.1016/j.jde.2008.08.004. |
[4] |
Arch. Ration. Mech. Anal., 159 (2001), 253-271.
doi: 10.1007/s002050100152. |
[5] |
J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
Boll. Unione Mat. Ital., 9 (2009), 93-104. |
[7] |
Cal. Var. Partial Differential Equations, 2 (1994), 29-48.
doi: 10.1007/BF01234314. |
[8] |
Comm. Math. Phys., 79 (1981), 167-180. |
[9] |
Ann. I. H. Poincaré-AN, 26 (2009), 943-958.
doi: 10.1016/j.anihpc.2008.03.009. |
[10] |
Topol. Methods Nonlinear Anal., 10 (1997), 1-13. |
[11] |
Commun. Appl. Anal., 7 (2003), 417-423. |
[12] |
Comm. Partial Differential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[13] |
Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[14] |
Advanced Nonlinear Studies, 8 (2008), 573-595. |
[15] |
Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[16] |
Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[17] |
Ann. Inst. H. Poincare Anal. Non Linéaire, 1 (1984), 223-283. |
[18] |
J. Funct. Anal., 149 (1997), 245-265. |
[19] |
Journ. Func. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[20] |
Math. Models Methods Appl. Sci., 15 (2005), 141-164. |
[21] |
J. Statist. Phys., 114 (2004), 179-204.
doi: 10.1023/B:JOSS.0000003109.97208.53. |
[22] |
Nonlinear Analysis, 71 (2009), 730-739.
doi: 10.1016/j.na.2008.10.105. |
[23] |
J. Differential Equations, 247 (2009), 618-647.
doi: 10.1016/j.jde.2009.03.002. |
[24] |
Nonlinear Analysis, 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
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