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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

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.

Citation: Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  Google Scholar

[3]

L. BergeA. 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.  Google Scholar

[4]

J. ByeonH. 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.  Google Scholar

[5]

J. ByeonH. 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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

L. BergeA. 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.  Google Scholar

[4]

J. ByeonH. 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.  Google Scholar

[5]

J. ByeonH. 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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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