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Scattering for a nonlinear Schrödinger equation with a potential

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  • We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global well-posedness analogous to the homogeneous case $V=0$ [10, 5]. Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.
    Mathematics Subject Classification: 35Q41, 35Q55, 35L15, 35B40, 35B20.


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  • [1]

    A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.doi: 10.1007/s002050050067.


    M. Beceanu and M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012), 471-481.doi: 10.1007/s00220-012-1435-x.


    H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.doi: 10.1007/BF00250555.


    J. Colliander, M. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832.


    T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.doi: 10.4310/MRL.2008.v15.n6.a13.


    A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.doi: 10.1016/0022-1236(86)90096-0.


    D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.doi: 10.1142/S0219891605000361.


    P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.doi: 10.1051/cocv:1998107.


    T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828.doi: 10.1155/IMRN.2005.2815.


    J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.doi: 10.1007/s00220-008-0529-y.


    A. Ionescu and D. Jerison, On the absence of positive eigenvalues of Schrödinger operators with rough potentials, Geom. Funct. Anal., 13 (2003), 1029-1081.doi: 10.1007/s00039-003-0439-2.


    M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.


    C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.doi: 10.1007/s00222-006-0011-4.


    C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.doi: 10.1007/s11511-008-0031-6.


    S. Keraani, On the defect of compactness for Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.doi: 10.1006/jdeq.2000.3951.


    R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arxiv.org/abs/1208.4904.


    M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.doi: 10.1007/BF00251502.


    K. McLeod, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbbR^n$, II. Trans. Amer. Math. Soc., 339 (1993), 495-505.doi: 10.2307/2154282.


    K. McLeod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.doi: 10.1007/BF00275874.


    I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513.doi: 10.1007/s00222-003-0325-4.


    Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546.


    A. Sikora and J. Wright, Imaginary powers of Laplace operators, Proc. Amer. Math. Soc., 129 (2001), 1745-1754doi: 10.1090/S0002-9939-00-05754-3.


    M. Takeda, Gaussian bounds of heat kernels for Schröinger operators on Riemannian manifolds, Bull. Lond. Math. Soc., 39 (2007), 85-94.doi: 10.1112/blms/bdl016.

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