• Previous Article
    The Orlicz Minkowski problem involving $ 0 < p < 1 $: From one constant to an infinite interval
  • DCDS-S Home
  • This Issue
  • Next Article
    Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms
June  2021, 14(6): 1967-1981. doi: 10.3934/dcdss.2021008

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

* Corresponding author: Jianfu Yang

Received  August 2020 Revised  November 2020 Published  June 2021 Early access  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 $
.
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. 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

[2]

J. ByeonH. 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. CunhaP. d'AveniaA. 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

show all references

References:
[1]

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

[2]

J. ByeonH. 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. CunhaP. d'AveniaA. 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

[1]

Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119

[2]

Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1931-1944. doi: 10.3934/dcdss.2021016

[3]

Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5541-5570. doi: 10.3934/dcds.2020237

[4]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193

[5]

Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053

[6]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064

[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[8]

Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389

[9]

Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703

[10]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83

[11]

Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199

[12]

Chenglin Wang, Jian Zhang. Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021036

[13]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100

[14]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[15]

Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013

[16]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048

[17]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025

[18]

Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891

[19]

Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431

[20]

Chunhua Li. Decay of solutions for a system of nonlinear Schrödinger equations in 2D. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4265-4285. doi: 10.3934/dcds.2012.32.4265

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (151)
  • HTML views (173)
  • Cited by (0)

Other articles
by authors

[Back to Top]