September  2016, 15(5): 1571-1601. doi: 10.3934/cpaa.2016003

Scattering for a nonlinear Schrödinger equation with a potential

1. 

University of Texas at Austin

Received  October 2014 Revised  March 2016 Published  July 2016

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.
Citation: Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
References:
[1]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 140 (1997), 285. doi: 10.1007/s002050050067. Google Scholar

[2]

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

[3]

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

[4]

J. Colliander, M. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications,, \emph{Adv. Differential Equations}, 19 (2014), 805. Google Scholar

[5]

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

[6]

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

[7]

D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[8]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev,, \emph{ESAIM Control Optim. Calc. Var.}, 3 (1998), 213. doi: 10.1051/cocv:1998107. Google Scholar

[9]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, \emph{Int. Math. Res. Not.}, 46 (2005), 2815. doi: 10.1155/IMRN.2005.2815. Google Scholar

[10]

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

[11]

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

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955. Google Scholar

[13]

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,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4. Google Scholar

[14]

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

[15]

S. Keraani, On the defect of compactness for Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513. Google Scholar

[22]

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

[23]

M. Takeda, Gaussian bounds of heat kernels for Schröinger operators on Riemannian manifolds,, \emph{Bull. Lond. Math. Soc.}, 39 (2007), 85. doi: 10.1112/blms/bdl016. Google Scholar

show all references

References:
[1]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 140 (1997), 285. doi: 10.1007/s002050050067. Google Scholar

[2]

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

[3]

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

[4]

J. Colliander, M. Czubak and J. Lee, Interaction Morawetz estimate for the magnetic Schrödinger equation and applications,, \emph{Adv. Differential Equations}, 19 (2014), 805. Google Scholar

[5]

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

[6]

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

[7]

D. Foschi, Inhomogeneous Strichartz estimates,, \emph{J. Hyperbolic Differ. Equ.}, 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[8]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev,, \emph{ESAIM Control Optim. Calc. Var.}, 3 (1998), 213. doi: 10.1051/cocv:1998107. Google Scholar

[9]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, \emph{Int. Math. Res. Not.}, 46 (2005), 2815. doi: 10.1155/IMRN.2005.2815. Google Scholar

[10]

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

[11]

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

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955. Google Scholar

[13]

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,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4. Google Scholar

[14]

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

[15]

S. Keraani, On the defect of compactness for Strichartz estimates of the Schrödinger equations,, \emph{J. Differential Equations}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials,, \emph{Ann. Inst. Fourier (Grenoble)}, 45 (1995), 513. Google Scholar

[22]

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

[23]

M. Takeda, Gaussian bounds of heat kernels for Schröinger operators on Riemannian manifolds,, \emph{Bull. Lond. Math. Soc.}, 39 (2007), 85. doi: 10.1112/blms/bdl016. Google Scholar

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