# American Institute of Mathematical Sciences

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
 [1] Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803 [2] Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 [3] Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 [4] Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831 [5] Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183 [6] Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481 [7] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 [8] Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 [9] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [10] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 [11] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [12] Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129 [13] Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122 [14] Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 [15] Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 [16] Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure & Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 [17] Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 [18] Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831 [19] Divyang G. Bhimani. The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5923-5944. doi: 10.3934/dcds.2019259 [20] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

2018 Impact Factor: 0.925