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

Youyan Wan; Jinggang Tan

doi:10.3934/dcds.2017119

Article Contents

2017, Volume 37, Issue 5: 2765-2786. Doi: 10.3934/dcds.2017119

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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
* Corresponding author: J. Tan

Received: January 31, 2016

Revised: October 31, 2016

Published: May 2017

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.

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Abstract

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

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Mathematics Subject Classification: 35J50, 35J10.

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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.