THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL

doi:10.1007/s10473-022-0318-2

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 1125-1140.doi: 10.1007/s10473-022-0318-2

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL

谭金岚, 李勇勇, 唐春雷

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

收稿日期:2020-12-11 发布日期:2022-06-24
通讯作者: Chunlei TANG,E-mail:tangcl@swu.edu.cn E-mail:tangcl@swu.edu.cn
基金资助:
The third author was supported by National Natural Science Foundation of China (11971393).

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL

Jinlan TAN, Yongyong LI, Chunlei TANG

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

Received:2020-12-11 Published:2022-06-24
Contact: Chunlei TANG,E-mail:tangcl@swu.edu.cn E-mail:tangcl@swu.edu.cn
Supported by:
The third author was supported by National Natural Science Foundation of China (11971393).

摘要/Abstract

摘要： In this paper, we investigate a class of nonlinear Chern-Simons-Schrödinger systems with a steep well potential. By using variational methods, the mountain pass theorem and Nehari manifold methods, we prove the existence of a ground state solution for λ > 0 large enough. Furthermore, we verify the asymptotic behavior of ground state solutions as λ → +∞.

关键词: Chern-Simons-Schrödinger system, steep well potential, ground state solution, concentration

Abstract: In this paper, we investigate a class of nonlinear Chern-Simons-Schrödinger systems with a steep well potential. By using variational methods, the mountain pass theorem and Nehari manifold methods, we prove the existence of a ground state solution for λ > 0 large enough. Furthermore, we verify the asymptotic behavior of ground state solutions as λ → +∞.

Key words: Chern-Simons-Schrödinger system, steep well potential, ground state solution, concentration

中图分类号:

35A01

谭金岚, 李勇勇, 唐春雷. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL[J]. 数学物理学报(英文版), 2022, 42(3): 1125-1140.

Jinlan TAN, Yongyong LI, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL[J]. Acta mathematica scientia,Series B, 2022, 42(3): 1125-1140.

参考文献

[1] Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on R^N. Comm Partial Differential Equations, 1995, 20(10):1725-1741
[2] Bartsch T, Wang Z Q. Multiple positive solutions for a nonlinear Schrödinger equation. Z Angew Math Phys, 2000, 51(3):366-384
[3] Bergé L, Bouard A D, Saut J C. Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation. Nonlinearity, 1995, 8(2):235-253
[4] Byeon J, Huh H, Seok J. Standing waves of nonlinear Schrödinger equations with the gauge field. J Funct Anal, 2012, 263(6):1575-1608
[5] 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(2):1285-1316
[6] Chen S T, Zhang B L, Tang X H. Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons-Schrödinger system in H¹(R²). Nonlinear Anal, 2019, 185:68-96
[7] Chen Z, Tang X H, Zhang J. Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in R². Adv Nonlinear Anal, 2019, 9(1):1066-1091
[8] Cunha P L, D'Avenia P, Pomponio A, et al. A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity. Nonlinear Differential Equations Appl, 2015, 22(6):1831-1850
[9] Huh H. Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. J Math Phys, 2012, 53 (6):8pp
[10] Jackiw R, Pi S Y. Classical and quantal nonrelativistic Chern-Simons theory. Phys Rev, 1990, 42(10):3500-3513
[11] Jackiw R, Pi S Y. Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys Rev Lett, 1990, 64(25):2969-2972
[12] Jackiw R, Pi S Y. Self-dual Chern-Simons solitons. Progr Theoret Phys Suppl, 1992, 107:1-40
[13] Jeanjean L. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal, 1997, 28(10):1633-1659
[14] Jeanjean L. On the existence of bounded Palais-Smale sequences and application to Landesman-Lazer-type problem set on R^N. Proc Roy Soc Edinburgh Sect A, 1999, 129(4):787-809
[15] Ji C, Fang F. Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth. J Math Anal Appl, 2017, 450(1):578-591
[16] Jiang Y S, Pomponio A, Ruiz D. Standing waves for a gauged nonlinear Schrödinger equation with a vortex point. Commun Contemp Math, 2016, 18 (4):Article ID 1550074 20pp
[17] 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(2):711-731
[18] Li G B, Luo X. Normalized solutions for the Chern-Simons-Schrödinger equation in R². Ann Acad Sci Fenn Math, 2017, 42(1):405-428
[19] 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, 2021, 66(3):476-486
[20] Liu B, Simth P, Tataru D. Local wellposedness of Chern-Simons-Schrödinger. Int Math Res Not, 2014, 2014(23):6341-6398
[21] Pankov A, Bartsch T, Wang Z Q. Nonlinear Schrödinger equations with steep potential well. Commun Contemp Math, 2001, 3(4):549-569
[22] Pomponio A, Ruiz D. Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc Var Partial Differential Equations, 2015, 53(1/8):289-316
[23] Pomponio A, Ruiz D. A variational analysis of a gauged nonlinear Schrödinger equation. J Eur Math Soc, 2015, 17(6):1463-1486
[24] Seok J. Infinitely many standing waves for the nonlinear Chern-Simons-Schrödinger equations. Adv Math Phys, 2015, 2015:1-7
[25] Tang X H, Zhang J, Zhang W. Existence and concentration of solutions for the Chern-Simons-Schrödinger system with general nonlinearity. Results Math, 2017, 71(3/8):643-655
[26] Wan Y Y, Tan J G. Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition. J Math Anal Appl, 2014, 415(1):422-434
[27] Wan Y Y, Tan J G. Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems. Nonlinear Differential Equations Appl, 2017, 24(3):28
[28] Wan Y Y, Tan J G. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete Contin Dyn Syst, 2017, 37(5):2765-2786
[29] Weinstein M I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm Math Phys, 1983, 87(4):567-576
[30] Willem M. Minimax theorems. Boston:Birkhäuser, 1996
[31] Xia A. Existence, nonexistence and multiplicity results of a Chern-Simons-Schrödinger system. Acta Appl Math, 2020, 166:147-159
[32] Xie W, Chen C. Sign-changing solutions for the nonlinear Chern-Simons-Schrödinger equations. Appl Anal, 2020, 99(5):880-898
[33] Yuan J. Multiple normalized solutions of Chern-Simons-Schrödinger system. Nonlinear Differential Equations Appl, 2015, 22(6):1801-1816
[34] Zhang N, et al. Ground state solutions for the Chern-Simons-Schrödinger equations with general nonlinearity. Complex Var Elliptic Equ, 2020, 65(8):1394-1411

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL

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