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Solutions to Chern-Simons-Schrödinger systems with external potential
1. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
2. | School of Sciences, Nanchang Institute of Technology, Nanchang 330099, China |
$ \begin{equation} \left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right. \end{equation} $ |
$ V(x) $ |
$ D_{0} = \partial_{t}+i\lambda A_{0} $ |
$ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $ |
$ (x_1,x_2,t)\in \mathbb{R}^{2,1} $ |
$ \lambda $ |
$ V(x) $ |
$ \lim_{|x|\to\infty}V(x) = +\infty $ |
$ 2<p<4 $ |
$ \lambda^*>0 $ |
$ 0<\lambda<\lambda^* $ |
$ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $ |
$ \tilde\lambda>0 $ |
$ \lambda>\tilde\lambda $ |
$ p>4 $ |
$ x\cdot \nabla V(x)\geq 0 $ |
$ \lambda>0 $ |
References:
[1] |
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. |
[2] |
J. Byeon, H. Huh and J. Seok,
On standing waves with a vortex point of order $N$ for the non-linear Chern-Simons-Schrödinger equations, J. Differ. Equ., 261 (2016), 1285-1316.
doi: 10.1016/j.jde.2016.04.004. |
[3] |
P. L. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano,
A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonlinear Differ. Equ. Appl., 22 (2015), 1831-1850.
doi: 10.1007/s00030-015-0346-x. |
[4] |
V. Dunne, Self-Dual Chern-Simons Theories, Springer, New York, 1995.
doi: 10.1007/978-3-540-44777-1. |
[5] |
Y. He and G. Li,
Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766.
doi: 10.5186/aasfm.2015.4041. |
[6] |
H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 063702, 8 pp.
doi: 10.1063/1.4726192. |
[7] |
R. Jackiw and S.-Y. Pi,
Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.
doi: 10.1103/PhysRevD.42.3500. |
[8] |
R. Jackiw and S.-Y. Pi,
Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64 (1990), 2969-2972.
doi: 10.1103/PhysRevLett.64.2969. |
[9] |
R. Jackiw and S.-Y. Pi,
Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40.
doi: 10.1143/PTPS.107.1. |
[10] |
Y. Jiang and H. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[11] |
Y. Jiang and H. Zhou,
Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential, Sci. China Math., 57 (2014), 1163-1174.
doi: 10.1007/s11425-014-4790-6. |
[12] |
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. |
[13] |
A. Pomponio and D. Ruiz,
Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.
doi: 10.1007/s00526-014-0749-2. |
[14] |
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. |
[15] |
Y. Wan and J. Tan,
The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786.
doi: 10.3934/dcds.2017119. |
[16] |
M. Willem, Minimax Thorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
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. |
[2] |
J. Byeon, H. Huh and J. Seok,
On standing waves with a vortex point of order $N$ for the non-linear Chern-Simons-Schrödinger equations, J. Differ. Equ., 261 (2016), 1285-1316.
doi: 10.1016/j.jde.2016.04.004. |
[3] |
P. L. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano,
A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonlinear Differ. Equ. Appl., 22 (2015), 1831-1850.
doi: 10.1007/s00030-015-0346-x. |
[4] |
V. Dunne, Self-Dual Chern-Simons Theories, Springer, New York, 1995.
doi: 10.1007/978-3-540-44777-1. |
[5] |
Y. He and G. Li,
Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766.
doi: 10.5186/aasfm.2015.4041. |
[6] |
H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 063702, 8 pp.
doi: 10.1063/1.4726192. |
[7] |
R. Jackiw and S.-Y. Pi,
Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.
doi: 10.1103/PhysRevD.42.3500. |
[8] |
R. Jackiw and S.-Y. Pi,
Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64 (1990), 2969-2972.
doi: 10.1103/PhysRevLett.64.2969. |
[9] |
R. Jackiw and S.-Y. Pi,
Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40.
doi: 10.1143/PTPS.107.1. |
[10] |
Y. Jiang and H. Zhou,
Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[11] |
Y. Jiang and H. Zhou,
Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential, Sci. China Math., 57 (2014), 1163-1174.
doi: 10.1007/s11425-014-4790-6. |
[12] |
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. |
[13] |
A. Pomponio and D. Ruiz,
Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.
doi: 10.1007/s00526-014-0749-2. |
[14] |
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. |
[15] |
Y. Wan and J. Tan,
The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786.
doi: 10.3934/dcds.2017119. |
[16] |
M. Willem, Minimax Thorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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