Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions

Filip Ficek

Archivum Mathematicum (2023)

  • Volume: 059, Issue: 1, page 31-38
  • ISSN: 0044-8753

Abstract

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Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.

How to cite

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Ficek, Filip. "Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions." Archivum Mathematicum 059.1 (2023): 31-38. <http://eudml.org/doc/298983>.

@article{Ficek2023,
abstract = {Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.},
author = {Ficek, Filip},
journal = {Archivum Mathematicum},
keywords = {nonlinear Schrödinger equation; stationary solutions; supercritical dimensions; shooting method},
language = {eng},
number = {1},
pages = {31-38},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions},
url = {http://eudml.org/doc/298983},
volume = {059},
year = {2023},
}

TY - JOUR
AU - Ficek, Filip
TI - Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 059
IS - 1
SP - 31
EP - 38
AB - Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.
LA - eng
KW - nonlinear Schrödinger equation; stationary solutions; supercritical dimensions; shooting method
UR - http://eudml.org/doc/298983
ER -

References

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  1. Bizoń, P., 10.1007/s10714-014-1724-0, Gen. Relativity Gravitation 46 (2014), 14 pp., Art. 1724. (2014) MR3205859DOI10.1007/s10714-014-1724-0
  2. Bizoń, P., Evnin, O., Ficek, F., A nonrelativistic limit for AdS perturbations, JHEP 12 (113) (2018), 18 pp. (2018) MR3900619
  3. Bizoń, P., Ficek, F., Pelinovsky, D.E., Sobieszek, S., Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential, Nonlinear Anal. 210 (2021), 36 pp., Paper No. 112358. (2021) MR4249792
  4. Busca, J., Sirakov, B., 10.1006/jdeq.1999.3701, J. Differential Equations 63 (2000), 41–56. (2000) DOI10.1006/jdeq.1999.3701
  5. Choquard, P., Stubbe, J., Vuffray, M., Stationary solutions of the Schrödinger-Newton model – an ODE approach, Differential Integral Equations 21 (2008), 665–679. (2008) MR2479686
  6. Clapp, M., Szulkin, A., 10.1007/s13324-022-00673-x, Anal. Math. Phys. 12 (2022), 1–19. (2022) MR4396665DOI10.1007/s13324-022-00673-x
  7. Ficek, F., 10.1103/PhysRevD.103.104062, Phys. Rev. D 103 (2021), 13 pp., Paper No. 104062. (2021) MR4277036DOI10.1103/PhysRevD.103.104062
  8. Gallo, C., Pelinovsky, D., On the Thomas-Fermi ground state in a harmonic potential, Asymptot. Anal. 73 (2011), 53–96. (2011) MR2841225
  9. Hastings, P., McLeod, J.B., Classical Methods in Ordinary Differential Equations With Applications to Boundary Value Problems, AMS Providence, Rhode Island, 2012. (2012) MR2865597
  10. Li, Y., Ni, W., 10.1080/03605309308820960, Comm. Partial Differential Equations 18 (1993), 1043–1054. (1993) DOI10.1080/03605309308820960
  11. Pelinovsky, D.E., Wei, J., Wu, Y., Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities, arXiv:2207.10145 [math.AP]. 
  12. Selem, F.H., 10.1088/0951-7715/24/6/006, Nonlinearity 24 (2011), 1795–1819. (2011) MR2802311DOI10.1088/0951-7715/24/6/006
  13. Selem, F.H., Kikuchi, H., 10.1016/j.jmaa.2011.09.034, J. Math. Anal. Appl. 387 (2012), 746–754. (2012) MR2853141DOI10.1016/j.jmaa.2011.09.034
  14. Selem, F.H., Kikuchi, H., Wei, J., Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity, Discrete Contin. Dyn. Syst. 33 (2013), 4613–4626. (2013) MR3049094
  15. Smith, R.A., Asymptotic stability of x ' ' + a ( t ) x ' + x = 0 , Q. J. Math. 12 (1961), 123–126. (1961) 

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