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Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. (English) Zbl 1014.35087

The paper deals with the existence of standing waves for a class of nonlinear Schrödinger equations in \(\mathbb{R}^n\), with both an electric and magnetic field. Under suitable non-degeneracy assumptions on the critical points of an auxiliary function related to the electric field, the authors prove the existence and the multiplicity of complex-valued solutions in the semiclassical limit. A relevant result is that, in the semiclassical limit, the presence of a magnetic field produces a phase in the complex wave, but it does not influence the location of peaks of the modules of these waves.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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[1] Ambrosetti, A.; Badiale, M., Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Roy. Soc. Edinburgh A, 128, 1131-1161 (1998) · Zbl 0928.34029
[2] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140, 285-300 (1997) · Zbl 0896.35042
[3] Ambrosetti, A.; Malchiodi, A.; Secchi, S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Rational Mech. Anal., 159, 253-271 (2001) · Zbl 1040.35107
[4] Ambrosetti, A.; Berti, M., Homoclinics and complex dynamics in slowly oscillating systems, Discr. Cont. Dyn. Systems, 4, 3, 285-300 (1998)
[5] Bott, R., Nondegenerate critical manifolds, Ann. of Math., 60, 2, 248-261 (1954) · Zbl 0058.09101
[6] Brummelhuis, R., Exponential decay in the semiclassical limit for eigenfunctions of Schrödinger operators with magnetic fields and potentials which degenerate at infinity, Comm. Partial Differential Equations, 16, 1489-1502 (1991) · Zbl 0749.35023
[7] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Birkhäuser
[8] S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations, to appear; S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differential Equations, to appear · Zbl 1062.81056
[9] Cingolani, S.; Lazzo, M., Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10, 1-13 (1997) · Zbl 0903.35018
[10] Cingolani, S.; Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160, 118-138 (2000) · Zbl 0952.35043
[11] Cingolani, S.; Nolasco, M., Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh, 128, 1249-1260 (1998) · Zbl 0922.35158
[12] Conley, C. C., Isolated Invariant Sets and the Morse Index. Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series in Mathematics, 38 (1978), American Mathematical Society · Zbl 0397.34056
[13] Del Pino, M.; Felmer, P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. PDE, 4, 121-137 (1996) · Zbl 0844.35032
[14] Del Pino, M.; Felmer, P., Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, 15, 127-149 (1998) · Zbl 0901.35023
[15] Esteban, M.; Lions, P. L., Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, (PDE and Calculus of Variations, in Honor of E. De Giorgi (1990), Birkhäuser)
[16] M. Grossi, Some results on a class of nonlinear Schrödinger equations, Math. Z., to appear; M. Grossi, Some results on a class of nonlinear Schrödinger equations, Math. Z., to appear
[17] Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 21, 787-820 (1996) · Zbl 0857.35116
[18] Helffer, B., On spectral theory for Schrödinger operators with magnetic potentials, (Adv. Stud. Pure Math., 23 (1994)), 113-141 · Zbl 0816.35100
[19] Helffer, B., Semiclassical analysis for Schrödinger operator with magnetic wells, (Rauch, J.; Simon, B., Quasiclassical Methods. Quasiclassical Methods, The IMA Volumes in Mathematics and Its Applications, 95 (1997), Springer: Springer New York) · Zbl 0887.35131
[20] Helffer, B.; Sjöstrand, J., Effet tunnel pour l’équation de Schrödinger avec champ magnétique, Ann. Scuola Norm. Sup. Pisa, 14, 625-657 (1987) · Zbl 0699.35205
[21] Kurata, K., Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal., 41, 763-778 (2000) · Zbl 0993.35081
[22] Li, Y. Y., On a singularly perturbed elliptic equation, Adv. Differential Equations, 2, 955-980 (1997) · Zbl 1023.35500
[23] Oh, Y. G., Existence of semiclassical bound states of nonlinear Schrödinger equations, Comm. Partial Differential Equations, 13, 1499-1519 (1988) · Zbl 0702.35228
[24] Wang, X.; Zeng, B., On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions, SIAM J. Math. Anal., 28, 633-655 (1997) · Zbl 0879.35053
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