Geodesic
Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions
Archivum mathematicum, Tome 59 (2023) no. 1, pp. 31-38 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2023-1-31
Classification : 34B15, 34B18, 35Q55
Keywords: nonlinear Schrödinger equation; stationary solutions; supercritical dimensions; shooting method 
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Ficek, Filip. Stationary solutions of semilinear Schrödinger equations with trapping potentials in supercritical dimensions. Archivum mathematicum, Tome 59 (2023) no. 1, pp. 31-38. doi: 10.5817/AM2023-1-31

[1] Bizoń, P.: Is AdS stable?. Gen. Relativity Gravitation 46 (2014), 14 pp., Art. 1724. | DOI | MR

[2] Bizoń, P., Evnin, O., Ficek, F.: A nonrelativistic limit for AdS perturbations. JHEP 12 (113) (2018), 18 pp. | MR

[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. | MR

[4] Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Differential Equations 63 (2000), 41–56. | DOI

[5] Choquard, P., Stubbe, J., Vuffray, M.: Stationary solutions of the Schrödinger-Newton model – an ODE approach. Differential Integral Equations 21 (2008), 665–679. | MR

[6] Clapp, M., Szulkin, A.: Non-variational weakly coupled elliptic systems. Anal. Math. Phys. 12 (2022), 1–19. | DOI | MR

[7] Ficek, F.: Schrödinger-Newton-Hooke system in higher dimensions: Stationary states. Phys. Rev. D 103 (2021), 13 pp., Paper No. 104062. | DOI | MR

[8] Gallo, C., Pelinovsky, D.: On the Thomas-Fermi ground state in a harmonic potential. Asymptot. Anal. 73 (2011), 53–96. | MR

[9] Hastings, P., McLeod, J.B.: Classical Methods in Ordinary Differential Equations With Applications to Boundary Value Problems. AMS Providence, Rhode Island, 2012. | MR

[10] Li, Y., Ni, W.: Radial symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$. Comm. Partial Differential Equations 18 (1993), 1043–1054. | DOI

[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.: Radial solutions with prescribed numbers of zeros for the nonlinear Schrödinger equation with harmonic potential. Nonlinearity 24 (2011), 1795–1819. | DOI | MR

[13] Selem, F.H., Kikuchi, H.: Existence and non-existence of solution for semilinear elliptic equation with harmonic potential and Sobolev critical/supercritical nonlinearities. J. Math. Anal. Appl. 387 (2012), 746–754. | DOI | MR

[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. | MR

[15] Smith, R.A.: Asymptotic stability of $x^{\prime \prime }+a(t)x^{\prime }+x=0$. Q. J. Math. 12 (1961), 123–126.

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