# American Institute of Mathematical Sciences

• Previous Article
Dimension reduction of thermistor models for large-area organic light-emitting diodes
• DCDS-S Home
• This Issue
• Next Article
Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation

## 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  January 2021

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrödinger system
 $$$\left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right.$$$
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 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 $. Citation: Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021008 ##### References:  [1] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608. doi: 10.1016/j.jfa.2012.05.024. Google Scholar [2] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order$N$for the non-linear Chern-Simons-Schrödinger equations, J. Differ. Equ., 261 (2016), 1285-1316. doi: 10.1016/j.jde.2016.04.004. Google Scholar [3] P. L. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano, A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonlinear Differ. Equ. Appl., 22 (2015), 1831-1850. doi: 10.1007/s00030-015-0346-x. Google Scholar [4] V. Dunne, Self-Dual Chern-Simons Theories, Springer, New York, 1995. doi: 10.1007/978-3-540-44777-1. Google Scholar [5] Y. He and G. Li, Standing waves for a class of Schrödinger-Poisson equations in$\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)$condition, J. Math. Anal. Appl., 415 (2014), 422-434. doi: 10.1016/j.jmaa.2014.01.084. Google Scholar [15] Y. Wan and J. Tan, The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786. doi: 10.3934/dcds.2017119. Google Scholar [16] M. Willem, Minimax Thorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar show all references ##### References:  [1] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608. doi: 10.1016/j.jfa.2012.05.024. Google Scholar [2] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order$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)$condition, J. Math. Anal. Appl., 415 (2014), 422-434. doi: 10.1016/j.jmaa.2014.01.084. Google Scholar [15] Y. Wan and J. Tan, The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786. doi: 10.3934/dcds.2017119. Google Scholar [16] M. Willem, Minimax Thorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar  [1] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 [2] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [3] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [4] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in$ \mathbb{R}^2 \$. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [5] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [6] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 [7] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [8] Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021 [9] Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 [10] Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 [11] Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 [12] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [13] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [14] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 [15] Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 [16] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [17] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [18] Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 [19] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [20] Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

2019 Impact Factor: 1.233