Abstract
In this paper we demonstrate the existence and concentration behavior of semi-classical solutions for the nonlinear Chern–Simons–Schrödinger systems with external potential. Combining the variational methods with concentration compactness principle, we prove the existence of a family of semi-classical solutions concentrating at the minimum points of the external potential.
Article PDF
Similar content being viewed by others
References
Berge, L., De Bouard, A., Saut, J.-C.: Blowing up time-dependent solutions of the planar, Chern–Simons gauged nonlinear Schrödinger equation. Nonlinearity 8, 235–253 (1995)
Byeon, J., Huh, H., Seok, J.: Standing waves of nonlinear Schrödinger equations with the gauge field. J. Funct. Anal. 263, 1575–1608 (2012)
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)
Deser, S., Jackiw, R., Templeton, S.: Topologically massive gauge theories. Ann. Phys. 140, 372–411 (1982)
Dunne, V.: Self-Dual Chern–Simons Theories. Springer, New York (1995)
Huh, H.: Standing waves of the Schrödinger equation coupled with the Chern–Simons gauge field. J. Math. Phys. 53, 063702 (2012)
Huh, H.: Nonexistence results of semilinear elliptic equations coupled the the Chern–Simons gauge field. Abstr. Appl. Anal. 2013, 1–5 (2013)
Hagen, C.: A new gauge theory without an elementary photon. Ann. Phys. 157, 342–359 (1984)
Jackiw, R., Pi, S.-Y.: Classical and quantal nonrelativistic Chern–Simons theory. Phys. Rev. D 42, 3500–3513 (1990)
Jackiw, R., Pi, S.-Y.: Self-dual Chern–Simons solitons. Prog. Theor. Phys. Suppl. 107, 1–40 (1992)
Jiang, Y., Pomponio, A., Ruiz, D.: Standing waves for a gauged nonlinear Schrödinger equation with a vortex point. Commun. Contemp. 18, 1550074 (2016)
Lions, P.L.: The concentration-compactness principle in the calculus of variation. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223–283 (1984)
Liu, B., Smith, P., Tataru, D.: Local wellposedness of Chern–Simons–Schrödinger. Int. Math. Res. Notices. doi:10.1093/imrn/rnt161
Pomponio, A., Ruiz, D.: A variational analysis of a gauged nonlinear Schrödinger equation. J. Eur. Math. Soc. 17, 1463–1486 (2015)
Pomponio, A., Ruiz, D.: Boundary concentration of a gauged nonlinear Schrödinger equation on large balls. Calc. Var. PDEs 53, 289–316 (2015)
Struwe, M.: Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (1996)
Tang, X., Zhang, J., Zhang, W.: Existence and concentration of solutions for the Chern–Simons–Schrödinger system with general nonlinearity. Results Math. 71(3), 643–655 (2017)
Wan, Y., Tan, J.: Standing waves for the Chern–Simons–Schrödinger systems without (AR) condition. J. Math. Anal. Appl. 415, 422–434 (2014)
Wan, Y., Tan, J.: The existence of nontrivial solutions to Chern–Simons–Schrödinger systems. Discrete Contin. Dyn. Syst. Ser. A (2017). doi:10.3934/dcds.2017119
Wang, X., Zeng, B.: On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J. Math. Anal. 28, 633–655 (1997)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Yuan, J.: Multiple normalized solutions of Chern–Simons–Schrödinger system. NoDEA Nonlinear Differ. Equ. Appl. 22, 1801–1816 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wan, Y., Tan, J. Concentration of semi-classical solutions to the Chern–Simons–Schrödinger systems. Nonlinear Differ. Equ. Appl. 24, 28 (2017). https://doi.org/10.1007/s00030-017-0448-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0448-8