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Topological pressure for the completely irregular set of birkhoff averages
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 |
We show the existence of nontrivial solutions to Chern-Simons-Schrödinger systems by using the concentration compactness principle and the argument of global compactness.
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] | |
[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] | |
[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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