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Standing waves of nonlinear Schrödinger equations with the gauge field. (English) Zbl 1248.35193
Summary: We study standing waves for nonlinear Schrödinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field A μ in terms of complex scalar field ϕ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35A15Variational methods (PDE)
35A01Existence problems for PDE: global existence, local existence, non-existence
References:
[1]Ambrosetti, A.: On Schrödinger-Poisson systems, Milan J. Math. 76, 257-274 (2008) · Zbl 1181.35257 · doi:10.1007/s00032-008-0094-z
[2]Ambrosetti, A.; Rabinowitz, P. H.: Dual variational methods in critical point theory and applications, J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[3]Benci, V.; Fortunato, D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. math. Phys. 14, No. 4, 409-420 (2002) · Zbl 1037.35075 · doi:10.1142/S0129055X02001168
[4]Berestycki, H.; Lions, P. -L.: Nonlinear scalar field equations, I. Existence of a ground state, Arch. ration. Mech. anal. 82, No. 4, 313-345 (1983) · Zbl 0533.35029
[5]Byeon, J.: Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. amer. Math. soc. 362, 1981-2001 (2010) · Zbl 1188.35082 · doi:10.1090/S0002-9947-09-04746-1
[6]D’aprile, T.; Mugnai, D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. roy. Soc. Edinburgh sect. A 134, No. 5, 893-906 (2004) · Zbl 1064.35182 · doi:10.1017/S030821050000353X
[7]Dunne, G. V.: Self-dual Chern-Simons theories, (1995)
[8]Gilbarg, D.; Trudinger, N.: Elliptic partial differential equations of second order, Grundlehren math. Wiss. 224 (1983) · Zbl 0562.35001
[9]Huh, H.: Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity 22, No. 5, 967-974 (2009)
[10]Jackiw, R.; Pi, S. -Y.: Classical and quantal nonrelativistic Chern-Simons theory, Phys. rev. D 42, 3500-3513 (1990)
[11]Jackiw, R.; Pi, S. -Y.: Self-dual Chern-Simons solitons, Progr. theoret. Phys. suppl. 107, 1-40 (1992)
[12]Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. funct. Anal. 237, 655-674 (2006) · Zbl 1136.35037 · doi:10.1016/j.jfa.2006.04.005