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June  2021, 14(6): 1967-1981. doi: 10.3934/dcdss.2021008

Solutions to Chern-Simons-Schrödinger systems with external potential

Lingyu Li 1, , Jianfu Yang 1,, and  Jinge Yang 2,

1. 

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
2. 

School of Sciences, Nanchang Institute of Technology, Nanchang 330099, China

* Corresponding author: Jianfu Yang

Received  August 2020 Revised  November 2020 Published  January 2021

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system
$ \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} $
with an external potential
$ V(x) $
, where
$ D_{0} = \partial_{t}+i\lambda A_{0} $
 and
$ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $
 for
$ (x_1,x_2,t)\in \mathbb{R}^{2,1} $
 are covariant derivatives,
$ \lambda $
 is the coupling number. Suppose that
$ V(x) $
 satisfies
$ \lim_{|x|\to\infty}V(x) = +\infty $
, we show for
$ 2<p<4 $
 that there exists
$ \lambda^*>0 $
 such that if
$ 0<\lambda<\lambda^* $
, problem (1) has two nontrivial static solutions
$ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $
. Moreover, there also exists
$ \tilde\lambda>0 $
 such that if
$ \lambda>\tilde\lambda $
, problem (1) has no nontrivial solutions. While for
$ p>4 $
 we assume in addition that
$ x\cdot \nabla V(x)\geq 0 $
, then problem (1) admits a mountain pass solution for all
$ \lambda>0 $
.
Keywords: Chern-Simons-Schrödinger system, variational method, nontrivial solutions.
Mathematics Subject Classification: Primary: 35J50; Secondary: 35J10.
Citation: Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1967-1981. doi: 10.3934/dcdss.2021008
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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