# American Institute of Mathematical Sciences

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

## 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

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.

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. [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. [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. [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. [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. [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. [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. [8] R. Benguria and H. Brezis, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules,, Comm. Math. Phys., 79 (1981), 167. [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. [10] S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1. [11] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, Commun. Appl. Anal., 7 (2003), 417. [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. [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. [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. [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. [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. [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. [18] M. Del Pino and P. Felmer, Semiclassical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245. [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. [20] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [8] R. Benguria and H. Brezis, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules,, Comm. Math. Phys., 79 (1981), 167. [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. [10] S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, Topol. Methods Nonlinear Anal., 10 (1997), 1. [11] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, Commun. Appl. Anal., 7 (2003), 417. [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. [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. [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. [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. [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. [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. [18] M. Del Pino and P. Felmer, Semiclassical states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245. [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. [20] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. [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. [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. [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. [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.
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