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Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation

  • *Corresponding author: Sangdon Jin

    *Corresponding author: Sangdon Jin
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  • For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [2]. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.

    Mathematics Subject Classification: Primary: 35J10; Secondary: 35J61.

    Citation:

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  • [1] J. BellazziniN. BoussaïdL. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.
    [2] J. BellazziniV. Georgiev and N. Visciglia, Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.  doi: 10.1007/s00208-018-1666-z.
    [3] J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp. doi: 10.1063/1.4726198.
    [4] T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.
    [5] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
    [6] T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.
    [7] W. ChoiY. Hong and J. Seok, On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1241-1263.  doi: 10.1017/prm.2018.114.
    [8] R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.
    [9] J. FröhlichB. L. G. Jonsson and E. Lenzmann, Effective dynamics for boson stars, Nonlinearity, 20 (2007), 1031-1075.  doi: 10.1088/0951-7715/20/5/001.
    [10] K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 473–478. doi: 10.3934/proc.2015.0473.
    [11] J. KriegerE. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.
    [12] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.
    [13] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.
    [14] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.
    [15] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684.
    [16] P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/s0294-1449(16)30422-x.
    [17] X. Luo and T. Yang, Stable solitary waves for pseudo-relativistic Hartree equations with short range potential, Nonlinear Anal., 207 (2021), Paper No. 112275, 13 pp. doi: 10.1016/j.na.2021.112275.
    [18] J. Seok and Y. Hong, Ground states to the generalized nonlinear schrödinger equations with bernstein symbols, Anal. Theory Appl., 37 (2021), 157-177.  doi: 10.4208/ata.2021.pr80.06.
    [19] Q. Shi and C. Peng, Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.
    [20] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.
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