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May  2017, 37(5): 2765-2786. doi: 10.3934/dcds.2017119

The existence of nontrivial solutions to Chern-Simons-Schrödinger systems

Youyan Wan 1, and  Jinggang Tan 2,,

1. 

The Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China
2. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile

* Corresponding author: J. Tan

Received  February 2016 Revised  November 2016 Published  February 2017

Fund Project: Y.W. was supported by Scientific Research Program of Hubei Provincial Department of Education(B2016299). J.T. was supported by Chile Government grant Fondecyt 1120105, Fondecyt 1160519, Proy. USM 121402, 121568; Spain Government grant MTM2011-27739-C04-01.

We show the existence of nontrivial solutions to Chern-Simons-Schrödinger systems by using the concentration compactness principle and the argument of global compactness.

Keywords: Chern-Simons gauge field, Schrödinger equation, variational methods.
Mathematics Subject Classification: 35J50, 35J10.
Citation: Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. 

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Archi. Rati. Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[3]

L. Berge, A. De Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007.

[4]

J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.

[5]

J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations, Journal of Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004.

[6]

P.L. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano, A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonl. Diff. Equ. Appl., 22 (2015), 1831-1850. 

[7]

W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/978-3-642-83743-2_2.

[8]

V. Dunne, Self-dual Chern-Simons Theories Springer, 1995. doi: 10.1007/978-3-540-44777-1.

[9]

H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field J. Math. Phys. 53 (2012), 063702, 8pp. doi: 10.1063/1.4726192.

[10]

H. Huh, Nonexistence results of semilinear elliptic equations coupled the the Chern-Simons gauge field Abstr. Appl. Anal. (2013), Art. ID 467985, 5 pp. doi: 10.1155/2013/467985.

[11]

R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513. 

[12]

R. Jackiw and S.-Y. Pi, Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40. 

[13]

Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrö dinger equation with a vortex point, Communications in Contemporary Mathematics 18 (2016), 1550074, 20pp.

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. 

[15]

B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, International Mathematics Research Notices, 23 (2014), 6341-6398.  doi: 10.1093/imrn/rnt161.

[16]

A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535.

[17]

A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Vari. PDEs, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2.

[18]

M. Struwe, Variational Methods Springer-Verlag, 1996.

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Zeit., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[20]

Y. Wan and J. Tan, Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition, J. Math. Anal. Appl., 415 (2014), 422-434.  doi: 10.1016/j.jmaa.2014.01.084.

[21]

Y. Wan and J. Tan, Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems, Preprint.

[22]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.  doi: 10.1137/S0036141095290240.

[23]

J. Yuan, Multiple normalized solutions of Chern-Simons-Schrödinger system, Nonl. Diff. Equ.Appl, 22 (2015), 1801-1816.  doi: 10.1007/s00030-015-0344-z.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. 

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Archi. Rati. Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[3]

L. Berge, A. De Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007.

[4]

J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.

[5]

J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations, Journal of Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004.

[6]

P.L. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano, A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonl. Diff. Equ. Appl., 22 (2015), 1831-1850. 

[7]

W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.  doi: 10.1007/978-3-642-83743-2_2.

[8]

V. Dunne, Self-dual Chern-Simons Theories Springer, 1995. doi: 10.1007/978-3-540-44777-1.

[9]

H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field J. Math. Phys. 53 (2012), 063702, 8pp. doi: 10.1063/1.4726192.

[10]

H. Huh, Nonexistence results of semilinear elliptic equations coupled the the Chern-Simons gauge field Abstr. Appl. Anal. (2013), Art. ID 467985, 5 pp. doi: 10.1155/2013/467985.

[11]

R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513. 

[12]

R. Jackiw and S.-Y. Pi, Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40. 

[13]

Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrö dinger equation with a vortex point, Communications in Contemporary Mathematics 18 (2016), 1550074, 20pp.

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. 

[15]

B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, International Mathematics Research Notices, 23 (2014), 6341-6398.  doi: 10.1093/imrn/rnt161.

[16]

A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535.

[17]

A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Vari. PDEs, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2.

[18]

M. Struwe, Variational Methods Springer-Verlag, 1996.

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Zeit., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[20]

Y. Wan and J. Tan, Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition, J. Math. Anal. Appl., 415 (2014), 422-434.  doi: 10.1016/j.jmaa.2014.01.084.

[21]

Y. Wan and J. Tan, Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems, Preprint.

[22]

X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.  doi: 10.1137/S0036141095290240.

[23]

J. Yuan, Multiple normalized solutions of Chern-Simons-Schrödinger system, Nonl. Diff. Equ.Appl, 22 (2015), 1801-1816.  doi: 10.1007/s00030-015-0344-z.

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