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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

* 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] Kazuhiro Kurata, Yuki Osada. 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