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The existence of ground state normalized solutions for Chern-Simons-Schrödinger systems

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Abstract

In this paper, we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H1(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)].

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References

  1. Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on ℝN. Comm Partial Differential Equations, 1995, 20: 1725–1741

    Article  MathSciNet  MATH  Google Scholar 

  2. Berestycki H, Lions P L. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch Rational Mech Anal, 1983, 82: 347–375

    Article  MathSciNet  MATH  Google Scholar 

  3. Byeon J, Huh H, Seok J. Standing waves of nonlinear Schrödinger equations with the gauge field. J Funct Anal, 2012, 263: 1575–1608

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen H B, **e W H. Existence and multiplicity of normalized solutions for the nonlinear Chern-Simons-Schrödinger equations. Ann Acad Sci Fenn Math, 2020, 45: 429–449

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen S, Zhang B, Tang X. Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons-Schrödinger system in H1(ℝ2). Nonlinear Anal, 2019, 185: 68–96

    Article  MathSciNet  MATH  Google Scholar 

  7. Gou T X, Zhang Z T. Normalized solutions to the Chern-Simons-Schrödinger system. J Funct Anal, 2021, 280: 10889

    Article  MATH  Google Scholar 

  8. Huh H. Blow-up solutions of the Chern-Simons-Schrödinger equations. Nonlinearity, 2009, 22: 967–974

    Article  MathSciNet  MATH  Google Scholar 

  9. Huh H. Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field. J Math Phys, 2012, 53: 063702

    Article  MathSciNet  MATH  Google Scholar 

  10. Jackiw R, Pi S-Y. Classical and quantal nonrelativistic Chern-Simons theory. Phys Rev D, 1990, 42(3): 3500–3513

    Article  MathSciNet  Google Scholar 

  11. Jackiw R, Pi S Y. Soliton solutions to the gauged nonlinear Schrödinger equation on the plane. Phys Rev Lett, 1990, 64: 2969–2972

    Article  MathSciNet  MATH  Google Scholar 

  12. Jeanjean L. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal, 1997, 28: 1633–1659

    Article  MathSciNet  MATH  Google Scholar 

  13. Ji C, Fang F. Standing waves for the Chern-Simons-Schröodinger equation with critical exponential growth, J Math Anal Appl, 2017, 450: 578–591

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  16. Li G, Luo X. Normalized solutions for the Chern-Simons-Schrödinger equation in ℝ2. Ann Acad Sci Fenn Math, 2017, 42: 405–428

    Article  MathSciNet  MATH  Google Scholar 

  17. Li G, Luo X, Shuai W. Sign-changing solutions to a gauged nonlinear Schröodinger equation. J Math Anal Appl, 2017, 455: 1559–1578

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  19. Li L, Yang J. Solutions to Chern-Simons-Schröodinger systems with erternal potential. Discret Contin Dyn Syst Ser S, 2021, 14: 1967–1981

    Google Scholar 

  20. Liang W, Zhai C. Existence of bound state solutions for the generalized Chern-Simons-Schröodinger system in H1(ℝ2). Appl Math Lett, 2020, 100: 106028

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu B, Smith P. Global wellposedness of the equivariant Chern-Simons-Schrödinger equation. Rev Mat Iberoam, 2016, 32: 751–794

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu B, Smith P, Tataru D. Local wellposedness of Chern-Simons-Schrödinger. Int Math Res Not, 2014, 2014(23): 6341–6398

    Article  MATH  Google Scholar 

  23. Luo X. Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation. Comput Math Appl, 2018, 76: 2701–2709

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  26. Pomponio A, Ruiz D. Boundary concentration of a gauged nonlinear Schroödinger equation on large balls. Calc Var Partial Differential Equations, 2015, 53: 289–316

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  28. Tang X, Zhang J, Zhang W. Existence and concentration of solutions for the Chern-Simons-Schröodinger system with general nonlinearity. Results Math, 2017, 71: 643–655

    Article  MathSciNet  MATH  Google Scholar 

  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-Schroödinger systems without (AR) condition. J Math Anal Appl, 2014, 415: 422–434

    Article  MathSciNet  MATH  Google Scholar 

  31. Wan Y, Tan J. Concentration of semi-classical solutions to the Chern-Simons-Schroödinger systems. NoDEA Nonlinear Differential Equations Appl, 2017, 24(3): Art 28

  32. Wan Y, Tan J. The existence of nontrivial solutions to Chern-Simons-Schröodinger systems. Discrete Contin Dyn Syst, 2017, 37: 2765–2786

    Article  MathSciNet  MATH  Google Scholar 

  33. Weinstein M I. Nonlinear Schröodinger equations and sharp interpolation estimates. Comm Math Phys, 1983, 87(4): 567–576

    Article  MATH  Google Scholar 

  34. Yuan J. Multiple normalized solutions of Chern-Simons-Schrödinger system. Nonlinear Differential Equations Appl, 2015, 22: 1801–1816

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang J, Zhang W, **e X. Infinitely many solutions for a gauged nonlinear Schrödinger equation. Appl Math Lett, 2019, 88: 21–27

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Yu Mao, ** Wu & Chunlei Tang

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  1. Yu Mao
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  2. ** Wu
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  3. Chunlei Tang
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Correspondence to ** Wu.

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Supported by the National Natural Science Foundation of China (11971393).

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Mao, Y., Wu, X. & Tang, C. The existence of ground state normalized solutions for Chern-Simons-Schrödinger systems. Acta Math Sci 43, 2649–2661 (2023). https://doi.org/10.1007/s10473-023-0620-7

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  • DOI: https://doi.org/10.1007/s10473-023-0620-7

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