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Chern-Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field. (English) Zbl 1304.35642

Summary: We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern-Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)-(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant \(q\) goes to infinity.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B06 Symmetries, invariants, etc. in context of PDEs
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[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[2] Benci, V.; Fortunato, D., Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14, 409-420 (2002) · Zbl 1037.35075
[3] Berge, L.; Bouard, A.; Saut, J. C., Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8, 235-253 (1995) · Zbl 0822.35125
[4] Bethuel, F.; Brezis, H.; Hélein, F., Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1, 123-148 (1993) · Zbl 0834.35014
[5] Byeon, J.; Huh, H.; Seok, J., Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263, 1575-1608 (2012) · Zbl 1248.35193
[6] Chae, D.; Imanuvilov, O. Y., Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems, J. Funct. Anal., 196, 87-118 (2002) · Zbl 1079.58009
[7] DʼAvenia, P., Non-radially symmetric solutions of nonlinear equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2, 177-192 (2002)
[8] Dunne, G. V.; Trugenberger, C. A., Self-duality and nonrelativistic Maxwell-Chern-Simons solitons, Phys. Rev. D, 34, 1323-1331 (1991)
[9] Han, J.; Jang, J., Nontopological bare solutions in the relativistic self-dual Maxwell-Chern-Simons-Higgs model, J. Math. Phys., 46, 1-16 (2005), Article No. 042310 · Zbl 1067.81094
[10] Han, J.; Kim, N., Nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains, J. Funct. Anal.. J. Funct. Anal., J. Funct. Anal., 242, 674-204 (2007), (Corrigendum) · Zbl 1332.35310
[11] Han, J.; Song, K., On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains, J. Math. Anal. Appl., 350, 1-8 (2009) · Zbl 1162.35038
[12] Huh, H., Standing waves of the Schrödinger equation coupled with the Chern-Simons gauged field, J. Math. Phys., 53, 063702 (2012) · Zbl 1276.81053
[13] Huh, H., Energy solution of the Chern-Simons-Schrodinger equations, Abstr. Appl. Anal. (2013), Art. ID 590653, 7 pp · Zbl 1276.35138
[14] Jackiw, R.; Pi, S.-Y., Soliton solutions to the gauged nonlinear Schrödinger equations on the plane, Phys. Rev. Lett., 64, 2969-2972 (1990) · Zbl 1050.81526
[15] Kwong, M., Uniqueness of positive solutions of \(\Delta u - u + u^p = 0 \text{in} R^n\), Arch. Ration. Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032
[16] Lee, C.; Lee, K.; Min, H., Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B, 252, 79-83 (1990)
[17] Lieb, E.; Loss, M., Analysis, Grad. Stud. Math., vol. 14 (1997)
[18] Palais, R. S., Principle of symmetric criticality, Comm. Math. Phys., 69, 19-30 (1979) · Zbl 0417.58007
[19] Ricciardi, T., Asymptotics for Maxwell-Chern-Simons multivortices, Nonlinear Anal., 50, 1093-1106 (2002) · Zbl 1079.58504
[20] Ricciardi, T.; Tarantello, G., Vortices in the Maxwell-Chern-Simons theory, Comm. Pure Appl. Math., 53, 811-851 (2000) · Zbl 1029.35207
[21] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 655-674 (2006) · Zbl 1136.35037
[22] Spruck, J.; Yang, Y., Existence theorems for periodic non-relativistic Maxwell-Chern-Simons solitons, J. Differential Equations, 127, 571-589 (1996) · Zbl 0848.35110
[23] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[24] Tarantello, G., Vortex-condensations of a non-relativistic Maxwell-Chern-Simons theory, J. Differential Equations, 141, 295-309 (1997) · Zbl 0890.35040
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