带有规范场的非线性 Schrödinger 方程在不定位势下的驻波解——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1360-1373.

带有规范场的非线性 Schrödinger 方程在不定位势下的驻波解——献给邓引斌教授 70 寿辰

丁雪彤(), 黄文涛^*()

华东交通大学理学院南昌 330013

收稿日期:2025-12-19 修回日期:2026-03-01 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 黄文涛 E-mail:13576737379@163.com;wthuang1014@aliyun.com
作者简介:丁雪彤,E-mail: 13576737379@163.com
基金资助:
国家自然科学基金(12001198);江西省自然科学基金(20232BAB201009)

Standing Waves for a Gauged Nonlinear Schrödinger Equation with Indefinite Potential

Xuetong Ding(), Wentao Huang^*()

School of Science, East China Jiaotong University, Nanchang 330013

Received:2025-12-19 Revised:2026-03-01 Online:2026-08-26 Published:2026-06-10
Contact: Wentao Huang E-mail:13576737379@163.com;wthuang1014@aliyun.com
Supported by:
NSFC(12001198);Natural Science Foundation of Jiangxi Province(20232BAB201009)

摘要/Abstract

摘要：

该文主要考虑平面上带有 Chern-Simons 规范场的非线性 Schrödinger 方程驻波解的存在性. 与文献中大多数现有工作相比, 该文主要新颖之处在于允许 Schrödinger 算子 $-\Delta+V$ 的符号是不定的, 因此相应的变分泛函不满足山路几何结构. 通过局部环绕技巧和无穷维 Morse 理论, 作者得到了该问题的一个非平凡解. 此外, 在非线性项为奇函数的情况下, 作者通过喷泉定理建立了该问题无穷多高能量解的存在性.

关键词: Chern-Simons 规范场, 不定位势, 驻波解, 临界群

Abstract:

In this paper, we mainly study the existence of standing wave solutions for nonlinear Schrödinger equations with the Chern-Simons gauge field on the plane. Compared with most existing works in the literature, the main novelty of this paper lies in allowing the sign of the Schrödinger operator $-\Delta+V$ to be indefinite, so that the corresponding variational functional does not satisfy the mountain pass geometry. By using a local linking technique and the infinite-dimensional Morse theory, we obtain a nontrivial solution to the problem. Moreover, under the assumption that the nonlinearity is odd, we establish the existence of infinitely many high-energy solutions via the fountain theorem.

Key words: Chern-Simons gauge field, indefinite potential, standing waves, critical groups

中图分类号:

O175.23

丁雪彤, 黄文涛. 带有规范场的非线性 Schrödinger 方程在不定位势下的驻波解——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1360-1373.

Xuetong Ding, Wentao Huang. Standing Waves for a Gauged Nonlinear Schrödinger Equation with Indefinite Potential[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1360-1373.

参考文献 38

[1]	Bartsch T, Li S J. Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal, 1997, 28(3): 419-441 doi: 10.1016/0362-546X(95)00167-T
[2]	Bergé L, De Bouard A, Saut J C. Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation. Nonlinearity, 1995, 8(2): 235-253 doi: 10.1088/0951-7715/8/2/007
[3]	Byeon J, Huh H, Seok J. Standing waves of nonlinear Schrödinger equations with the gauge field. J Funct Anal, 2012, 263(6): 1575-1608 doi: 10.1016/j.jfa.2012.05.024
[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(2): 1285-1316 doi: 10.1016/j.jde.2016.04.004
[5]	Chang K C. Infiinite-Dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and their Applications. Boston: Birkhäauser, 1993
[6]	Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71(10): 4927-4934 doi: 10.1016/j.na.2009.03.050
[7]	Chen X W, Smith P. On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy. Anal PDE, 2014, 7(7): 1683-1712 doi: 10.2140/apde
[8]	Cunha P L, d'Avenia P, Pomponio A, Siciliano G. A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity. NoDEA Nonlinear Differ Equ Appl, 2015, 22(6): 1831-1850 doi: 10.1007/s00030-015-0346-x
[9]	Deng Y B, Peng S J, Shuai W. Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^2$. J Differential Equations, 2018, 264(6): 4006-4035 doi: 10.1016/j.jde.2017.12.003
[10]	Ding Y H, Szulkin A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc Var Partial Differential Equations, 2007, 29(3): 397-419 doi: 10.1007/s00526-006-0071-8
[11]	Ding Y H, Wei J C. Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J Funct Anal, 2007, 251(2): 546-572 doi: 10.1016/j.jfa.2007.07.005
[12]	Dunne G V. Self-Dual Chern-Simons Theories. New York: Springer, 1995
[13]	Huh H. Blow-up solutions of the Chern-Simons-Schrödinger equations. Nonlinearity, 2009, 22(5): 967-974 doi: 10.1088/0951-7715/22/5/003
[14]	Huh H. Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. J Math Phys, 2012, 53(6): Art 063702
[15]	Jackiw R, Pi S Y. Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys Rev Lett, 1990, 64(25): 2969-2972 pmid: 10041861
[16]	Jackiw R, Pi S Y. Classical and quantal nonrelativistic Chern-Simons theory. Phys Rev D, 1990, 42(10): 3500-3513 pmid: 10012752
[17]	Jackiw R, Pi S Y. Self-dual Chern-Simons solitons. Progr Theoret Phys Suppl, 1992, 107: 1-40 doi: 10.1143/PTPS.107.1
[18]	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): Art 1550074
[19]	Li S J, Willem M. Applications of local linking to critical point theory. J Math Anal Appl, 1995, 189(1): 6-32 doi: 10.1006/jmaa.1995.1002
[20]	Liu J Q. The Morse index of a saddle point. J Systems Sci Math Sci, 1989, 2(1): 32-39
[21]	Liu B P, Smith P. Global wellposedness of the equivariant Chern-Simons-Schrödinger equation. Rev Mat Iberoam, 2016, 32(3): 751-794 doi: 10.4171/rmi
[22]	Liu B P, Smith P, Tataru D. Local wellposedness of Chern-Simons-Schrödinger. Int Math Res Not, 2014, 2014(23): 6341-6398 doi: 10.1093/imrn/rnt161
[23]	Liu S B, Zhou J. Standing waves for quasilinear Schrödinger equations with indefinite potentials. J Differential Equations, 2018, 265(9): 3970-3987 doi: 10.1016/j.jde.2018.05.024
[24]	Luo X. Multiple normalized solutions for a planar gauged nonlinear Schrödinger equation. Z Angew Math Phys, 2018, 69(3): Art 58
[25]	Mawhin J, Willem M. Critical point theory and Hamiltonian systems. Applied Mathematical Sciences. New York: Springer, 1989
[26]	Niederer U. The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials. Helv Phys Acta, 1974, 47(2): 167-172
[27]	Pomponio A, Ruiz D. A variational analysis of a gauged nonlinear Schrödinger equation. J Eur Math Soc, 2015, 17(6): 1463-1486 doi: 10.4171/jems
[28]	Pomponio A, Ruiz D. Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc Var Partial Differential Equations, 2015, 53(1/2): 289-316 doi: 10.1007/s00526-014-0749-2
[29]	Pomponio A, Shen L J, Zeng X Y, Zhang Y M. Generalized Chern-Simons-Schrödinger system with signchanging steep potential well: Critical and subcritical exponential case. J Geom Anal, 2023, 33(6): Art 185
[30]	Shen L J, Squassina M. Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth. J Math Anal Appl, 2025, 543(2): Art 128926
[31]	Shen L J, Squassina M, Yang M B. Critical gauged Schrödinger equation in $\mathbb{R}^2$ with vanishing potentials. Discrete Contin Dyn Syst, 2022, 42(9): 4415-4438
[32]	Seok J. Infinitely many standing waves for the nonlinear Chern-Simons-Schrödinger equations. Adv Math Phys, 2015, 2015: Art 519374
[33]	Tan J L, Li Y Y, Tang C L. The existence and concentration of ground state solutions for Chern-Simons-Schrödinger systems with a steep well potential. Acta Math Sci, 2022, 42B(3): 1125-1140
[34]	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(3/4): 643-655 doi: 10.1007/s00025-016-0553-8
[35]	Wan Y Y, Tan J G. Concentration of semi-classical solutions to the Chern-Simons-Schrödinger system. NoDEA Nonlinear Differ Equ Appl, 2017, 24(3): Art 28
[36]	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 doi: 10.3934/dcds.2017119
[37]	Wang Z Q. On a superlinear elliptic equation. Ann Inst H Poincaré Anal Non Linéaire, 1991, 8(1): 43-57 doi: 10.4171/aihpc
[38]	Willem M. Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications. Boston: Birkhäuser, 1996