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Existence and Concentration of Semi-classical Ground State Solutions for Chern–Simons–Schrödinger System

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Abstract

In this paper, we consider the equation

$$\begin{aligned} -\varepsilon ^{2}\Delta u+ V(x)u+\left( A_{0}(u)+A_{1}^{2}(u)+A_{2}^{2}(u)\right) u=f(u) \ \ \ \ \mathrm {in} ~ H^{1}({\mathbb {R}}^{2}), \end{aligned}$$

where \(\varepsilon \) is a small parameter, V is the external potential, \(A_i(i=0,1,2)\) is the gauge field and \(f\in C({\mathbb {R}}, {\mathbb {R}})\) is 5-superlinear growth. By using variational methods and analytic technique, we prove that this system possesses a ground state solution \(u_\varepsilon \). Moreover, our results show that, as \(\varepsilon \rightarrow 0\), the global maximum point \(x_\varepsilon \) of \(u_\varepsilon \) must concentrate at the global minimum point \(x_0\) of V.

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References

  1. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations: I: existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

    Article  MathSciNet  Google Scholar 

  2. Bergé, L., de Bouard, A., Saut, J.-C.: Blowing up time-dependent solutions of the planar, Chern–Simons gauged nonlinear schrodinger equation. Nonlinearity 8, 235 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, S., Zhang, B., Tang, X.: Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in \(H^1(\mathbb{R}^2)\). Nonlinear Anal. 185, 68–96 (2019)

    Article  MathSciNet  Google Scholar 

  5. 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. 22, 1831–1850 (2015)

    Article  Google Scholar 

  6. Deng, Y., Peng, S., Shuai, W.: Nodal standing waves for a gauged nonlinear Schrödinger equation in \(\mathbb{R}^2\). J. Differ. Equ. 264, 4006–4035 (2018)

    Article  Google Scholar 

  7. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, (2001), Reprint of the 1998 edition

  8. Han, J., Huh, H., Seok, J.: Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field. J. Funct. Anal. 266, 318–342 (2014)

    Article  MathSciNet  Google Scholar 

  9. Huh, H.: Standing waves of the Schrödinger equation coupled with the Chern–Simons gauge field. J. Math. Phys. 53(8), 063702 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Jackiw, R., Pi, S.-Y.: Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 1–40, Low-dimensional field theories and condensed matter physics (Kyoto, 1991)

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

    Article  MathSciNet  Google Scholar 

  14. Li, G., Luo, X.: Normalized solutions for the Chern–Simons–Schrödinger equation in \({\mathbb{R}}^2\). Ann. Acad. Sci. Fenn. Math. 42, 405–428 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. 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). https://doi.org/10.1080/17476933.2020.1723564

    Article  MATH  Google Scholar 

  17. Liu, J., Liao, J.-F., Tang, C.-L.: A positive ground state solution for a class of asymptotically periodic Schrödinger equations. Comput. Math. Appl. 71, 965–976 (2016)

    Article  MathSciNet  Google Scholar 

  18. Liu, Z., Ouyang, Z., Zhang, J.: Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in \(\mathbb{R}^2\). Nonlinearity 32, 3082–3111 (2019)

    Article  MathSciNet  Google Scholar 

  19. Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

    Article  MathSciNet  Google Scholar 

  20. Seok, J.: Infinitely many standing waves for the nonlinear Chern–Simons–Schrödinger equations. Adv. Math. Phys. 519374, 7 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Wan, Y., Tan, J.: Concentration of semi-classical solutions to the Chern–Simons–Schrödinger systems. NoDEA Nonlinear Differ. Equ. Appl. 24, 24 (2017)

    Article  Google Scholar 

  22. Wan, Y., Tan, J.: The existence of nontrivial solutions to Chern–Simons–Schrödinger systems. Discrete Contin. Dyn. Syst. 37, 2765–2786 (2017)

    Article  MathSciNet  Google Scholar 

  23. Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston Inc, Boston, MA (1996)

  24. Xie, W., Chen, C.: Sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations. Appl. Anal. 99, 880–898 (2020)

    Article  MathSciNet  Google Scholar 

  25. Yuan, J.: Multiple normalized solutions of Chern–Simons–Schrödinger system. NoDEA Nonlinear Differ. Equ. Appl. 22, 1801–1816 (2015)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Lin-Jing Wang, Gui-Dong Li & Chun-Lei Tang

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  1. Lin-Jing Wang
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  2. Gui-Dong Li
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Supported by National Natural Science Foundation of China (No.11971393).

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Wang, LJ., Li, GD. & Tang, CL. Existence and Concentration of Semi-classical Ground State Solutions for Chern–Simons–Schrödinger System. Qual. Theory Dyn. Syst. 20, 40 (2021). https://doi.org/10.1007/s12346-021-00480-y

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