THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS*

doi:10.1007/s10473-023-0620-7

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2649-2661.doi: 10.1007/s10473-023-0620-7

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*

Yu MAO, Xingping WU^†, Chunlei TANG

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

收稿日期:2022-04-15 修回日期:2023-05-23 发布日期:2023-12-08
通讯作者: †Xingping Wu, E-mail: wuxp@swu.edu.cn
作者简介:Yu Mao, E-mail: 2531416750@qq.com; Chunlei Tang, E-mail: tangcl@swu.edu.cn
基金资助:
Supported by the National Natural Science Foundation of China (11971393).

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*

Yu MAO, Xingping WU^†, Chunlei TANG

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received:2022-04-15 Revised:2023-05-23 Published:2023-12-08
Contact: †Xingping Wu, E-mail: wuxp@swu.edu.cn
About author:Yu Mao, E-mail: 2531416750@qq.com; Chunlei Tang, E-mail: tangcl@swu.edu.cn
Supported by:
Supported by the National Natural Science Foundation of China (11971393).

摘要/Abstract

摘要： In this paper, we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in $H^{1}(\mathbb{R}^{2})$. When the nonlinearity satisfies some general 3-superlinear conditions, we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in [L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. (1997)].

关键词: Chern-Simons-Schrödinger system;, non-constant potential, Pohožaev identity;, ground state normalized solution

Abstract: In this paper, we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in $H^{1}(\mathbb{R}^{2})$. When the nonlinearity satisfies some general 3-superlinear conditions, we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in [L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. (1997)].

Key words: Chern-Simons-Schrödinger system;, non-constant potential, Pohožaev identity;, ground state normalized solution

中图分类号:

35A01

Yu MAO, Xingping WU, Chunlei TANG. THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(6): 2649-2661.

Yu MAO, Xingping WU, Chunlei TANG. THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2649-2661.

参考文献

[1] Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on ${ \mathbb{R}}^N$. Comm Partial Differential Equations, 1995, 20: 1725-1741
[2] Berestycki H, Lions P L. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch Rational Mech Anal, 1983, 82: 347-375
[3] Byeon J, Huh H, Seok J. Standing waves of nonlinear Schrödinger equations with the gauge field. J Funct Anal, 2012, 263: 1575-1608
[4] Byeon J, Huh H, Seok J. On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations. J Differential Equations, 2016, 261: 1285-1316
[5] Chen H B, Xie W H. Existence and multiplicity of normalized solutions for the nonlinear Chern-Simons-Schrödinger equations. Ann Acad Sci Fenn Math, 2020, 45: 429-449
[6] Chen S, Zhang B, Tang X. Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons-Schrödinger system in $H^1(\Bbb{R}^2)$. Nonlinear Anal, 2019, 185: 68-96
[7] Gou T X, Zhang Z T. Normalized solutions to the Chern-Simons-Schrödinger system. J Funct Anal, 2021, 280: 10889
[8] Huh H. Blow-up solutions of the Chern-Simons-Schrödinger equations. Nonlinearity, 2009, 22: 967-974
[9] Huh H. Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. J Math Phys, 2012, 53: 063702
[10] Jackiw R, Pi S -Y. Classical and quantal nonrelativistic Chern-Simons theory. Phys Rev D, 1990, 42(3): 3500-3513
[11] Jackiw R, Pi S Y. Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys Rev Lett, 1990, 64: 2969-2972
[12] Jeanjean L. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal, 1997, 28: 1633-1659
[13] Ji C, Fang F. Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth, J Math Anal Appl, 2017, 450: 578-591
[14] Kang J C, Li Y Y, Tang C L. Sign-changing solutions for Chern-Simons-Schrödinger equations with asymptotically 5-linear nonlinearity. Bull Malays Math Sci Soc, 2021, 44: 711-731
[15] Kang J C, Tang C L.Existence of nontrivial solutions to Chern-Simons-Schrödinger equations with indefinite potential. Discret Contin Dyn Syst Ser S, 2021, 14: 1931-1944
[16] Li G, Luo X. Normalized solutions for the Chern-Simons-Schrödinger equation in ${ \mathbb{R}}^2$. Ann Acad Sci Fenn Math, 2017, 42: 405-428
[17] Li G, Luo X, Shuai W. Sign-changing solutions to a gauged nonlinear Schrödinger equation. J Math Anal Appl, 2017, 455: 1559-1578
[18] Li G D, Li Y Y, Tang C L. Existence and concentrate behavior of positive solutions for Chern-Simons-Schrödinger systems with critical growth. Complex Var Elliptic Equ, 2020, 66(3): 476-486
[19] Li L, Yang J.Solutions to Chern-Simons-Schrödinger systems with erternal potential. Discret Contin Dyn Syst Ser S, 2021, 14: 1967-1981
[20] Liang W, Zhai C. Existence of bound state solutions for the generalized Chern-Simons-Schrödinger system in $H^1(\Bbb{R}^2)$. Appl Math Lett, 2020, 100: 106028
[21] Liu B, Smith P. Global wellposedness of the equivariant Chern-Simons-Schrödinger equation. Rev Mat Iberoam, 2016, 32: 751-794
[22] Liu B, Smith P, Tataru D. Local wellposedness of Chern-Simons-Schrödinger. Int Math Res Not, 2014, 2014(23): 6341-6398[23] Luo X. Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation. Comput Math Appl, 2018, 76: 2701-2709
[24] Luo X. Multiple normalized solutions for a planar gauged nonlinear Schrödinger equation. Z Angew Math Phys, 2018, 69: Art 58
[25] Mao Y, Wu X P, Tang C L. Existence and multiplicity of solutions for asymptotically 3-linear Chern-Simons-Schrödinger systems. J Math Anal Appl, 2021, 498: 124939
[26] Pomponio A, Ruiz D. Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc Var Partial Differential Equations, 2015, 53: 289-316
[27] Tan J, Li Y, Tang C. The existance and concentration of ground state solutions for Chern-Simons-Schrödinger system with a steep well potential. Acta Mathematica Scientia, 2022, 42B(3): 1125-1140
[28] Tang X, Zhang J, Zhang W. Existence and concentration of solutions for the Chern-Simons-Schrödinger system with general nonlinearity. Results Math, 2017, 71: 643-655
[29] Wang L J, Li G D, Tang C L. Existence and Concentration of Semi-classical Ground State Solutions for Chern-Simons-Schrödinger System. Qualitative Theory of Dynamical Systems, 2021, 20: Art 40
[30] Wan Y, Tan J. Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition. J Math Anal Appl, 2014, 415: 422-434
[31] Wan Y, Tan J. Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems. NoDEA Nonlinear Differential Equations Appl, 2017, 24(3): Art 28
[32] Wan Y, Tan J. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete Contin Dyn Syst, 2017, 37: 2765-2786
[33] Weinstein M I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm Math Phys, 1983, 87(4): 567-576
[34] Yuan J. Multiple normalized solutions of Chern-Simons-Schrödinger system. Nonlinear Differential Equations Appl, 2015, 22: 1801-1816
[35] Zhang J, Zhang W, Xie X. Infinitely many solutions for a gauged nonlinear Schrödinger equation. Appl Math Lett, 2019, 88: 21-27

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*

THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*

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