-
Previous Article
A p-Laplacian supercritical Neumann problem
- DCDS Home
- This Issue
-
Next Article
An approximation solvability method for nonlocal semilinear differential problems in Banach spaces
Scattering of solutions to the nonlinear Schrödinger equations with regular potentials
1. | College of Science, Hohai University, Nanjing 210098, Jiangsu, China |
2. | Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of Mathematic, University of Science and Technology of China, Hefei 230026, Anhui, China |
In this paper, we prove the scattering of radial solutions to high dimensional energy-critical nonlinear Schrödinger equations with regular potentials in the defocusing case.
References:
[1] |
P. Alsholm and G. Schmidt,
Spectral and scattering theory for Schrödinger operators, Arch. Rational Mech. Anal, 40 (1971), 281-311.
doi: 10.1007/BF00252679. |
[2] |
P. Antonelli, R. Carles and J. D. Silva,
Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.
doi: 10.1007/s00220-014-2166-y. |
[3] |
V. Banica and N. Visciglia,
Scattering for non linear Schrödinger equation with a delta potential, J. Differential Equations, 260 (2016), 4410-4439.
doi: 10.1016/j.jde.2015.11.016. |
[4] |
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. |
[5] |
J. Bourgain,
Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
doi: 10.1090/S0894-0347-99-00283-0. |
[6] |
P. Chen, J. Magniez and E. M. Ouhabaz,
Riesz transforms on non-compact manifolds, arXiv: 1411.0137. |
[7] |
J. Colliander, M. Czubak and J. Lee,
Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832.
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[9] |
S. Cuccagna, V. Georgiev and N. Visciglia,
Decay and scattering of small solutions of pure power NLS in $\mathbb{R}$ with p>3 and with a potential, Comm. Pure Appl. Math., 67 (2014), 957-981.
doi: 10.1002/cpa.21465. |
[10] |
P. D'ancona, L. Fanelli, L. Vega and N. Visciglia,
Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.
doi: 10.1016/j.jfa.2010.02.007. |
[11] |
P. D'ancona and V. Pierfelice,
On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.
doi: 10.1016/j.jfa.2005.05.013. |
[12] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.
doi: 10.1090/S0894-0347-2011-00727-3. |
[13] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), no. 18,3435-3516.
doi: 10.1090/S0894-0347-2011-00727-3. |
[14] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 to appear in
Amer. J. Math. , arXiv: 1010.0040.
doi: 10.1090/S0894-0347-2011-00727-3. |
[15] |
B. Dodson,
Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Advances in Mathematics, 285 (2015), 1589-1618.
doi: 10.1016/j.aim.2015.04.030. |
[16] |
B. Dodson,
Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold,
arXiv: 1409. 1950. |
[17] |
J. Ginibre, T. Ozawa and G. Velo,
On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor., 60 (1994), 211-239.
|
[18] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl.(9), 64 (1985), 363-401.
|
[19] |
R. H. Goodman, R. E. Slusher and M. I. Weinstein,
Stopping light on a defect, J. Opt. Soc. Am. B, 19 (2002), 1635-1652.
doi: 10.1364/JOSAB.19.001635. |
[20] |
R. H. Goodman, M. I. Weinstein and P. J. Holmes,
Nonlinear propagation of light in one-dimensional periodic structures, J. Nonlinear Sci., 11 (2001), 123-168.
doi: 10.1007/s00332-001-0002-y. |
[21] |
Z. Hani and L. Thomann,
Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. Pure Appl. Math., 69 (2016), 1727-1776.
doi: 10.1002/cpa.21594. |
[22] |
K. Hepp,
The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277.
doi: 10.1007/BF01646348. |
[23] |
Y. Hong,
Scattering for a nonlinear Schrödinger equation with a potential, Comm. Pure Appl. Anal.(5), 15 (2016), 1571-1601.
doi: 10.3934/cpaa.2016003. |
[24] |
S. Ibrahim, N. Masmoudi and K. Nakanishi,
Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.
doi: 10.2140/apde.2011.4.405. |
[25] |
J. L. Journe, A. Soffer and C. D. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[26] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[27] |
S. Keraani,
On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[28] |
C. E. 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. |
[29] |
R. Killip and M. Visan,
The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.
doi: 10.1353/ajm.0.0107. |
[30] |
R. Killip, M. Visan and X. Zhang,
The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Analysis and PDE, 1 (2009), 229-266.
doi: 10.2140/apde.2008.1.229. |
[31] |
D. Lafontaine,
Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39.
doi: 10.3233/ASY-161384. |
[32] |
E. H. Lieb, R. Seiringer and J. Yngvason,
A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.
doi: 10.1007/s002200100533. |
[33] |
H. P. McKean and J. Shatah,
The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math., 44 (1991), 1067-1080.
doi: 10.1002/cpa.3160440817. |
[34] |
K. Nakanishi,
Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, Journal of Functional Analysis, 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[35] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290.
|
[36] |
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. |
[37] |
E. Ryckman and M. Visan,
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
doi: 10.1353/ajm.2007.0004. |
[38] |
W. Schlag,
Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285.
|
[39] |
A. Soffer and M. I. Weinstein,
Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.
doi: 10.1007/s002220050303. |
[40] |
H. Spohn,
Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.
doi: 10.1103/RevModPhys.52.569. |
[41] |
W. Strauss, Nonlinear scattering theory. Scattering theory in mathematical physics, Proceedings of the NATO Advanced Study Institue, (Denver, 1973), 53-78. NATO Advanced Science Institues, Volume C9. Reidel, Dordrecht, 1974. |
[42] |
W. Strauss,
Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.
doi: 10.1016/0022-1236(81)90019-7. |
[43] |
T. Tao,
Nonlinear Dispersive Equations: Local and Global Analysis
American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
[44] |
T. Tao, M. Visan and X. Zhang,
Minimal-mass blowup solutions of the mass-critical NLS, Forum Mathematicum, 20 (2008), 881-919.
doi: 10.1515/FORUM.2008.042. |
[45] |
T. Tao, M. Visan and X. Zhang,
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math J., 140 (2007), 165-202.
doi: 10.1215/S0012-7094-07-14015-8. |
[46] |
M. C. Vilela,
Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
[47] |
N. Visciglia,
On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.
doi: 10.4310/MRL.2009.v16.n5.a14. |
[48] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimemsions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
show all references
References:
[1] |
P. Alsholm and G. Schmidt,
Spectral and scattering theory for Schrödinger operators, Arch. Rational Mech. Anal, 40 (1971), 281-311.
doi: 10.1007/BF00252679. |
[2] |
P. Antonelli, R. Carles and J. D. Silva,
Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys., 334 (2015), 367-396.
doi: 10.1007/s00220-014-2166-y. |
[3] |
V. Banica and N. Visciglia,
Scattering for non linear Schrödinger equation with a delta potential, J. Differential Equations, 260 (2016), 4410-4439.
doi: 10.1016/j.jde.2015.11.016. |
[4] |
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. |
[5] |
J. Bourgain,
Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
doi: 10.1090/S0894-0347-99-00283-0. |
[6] |
P. Chen, J. Magniez and E. M. Ouhabaz,
Riesz transforms on non-compact manifolds, arXiv: 1411.0137. |
[7] |
J. Colliander, M. Czubak and J. Lee,
Interaction Morawetz estimate for the magnetic Schrödinger equation and applications, Adv. Differential Equations, 19 (2014), 805-832.
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math., 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[9] |
S. Cuccagna, V. Georgiev and N. Visciglia,
Decay and scattering of small solutions of pure power NLS in $\mathbb{R}$ with p>3 and with a potential, Comm. Pure Appl. Math., 67 (2014), 957-981.
doi: 10.1002/cpa.21465. |
[10] |
P. D'ancona, L. Fanelli, L. Vega and N. Visciglia,
Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.
doi: 10.1016/j.jfa.2010.02.007. |
[11] |
P. D'ancona and V. Pierfelice,
On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.
doi: 10.1016/j.jfa.2005.05.013. |
[12] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.
doi: 10.1090/S0894-0347-2011-00727-3. |
[13] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), no. 18,3435-3516.
doi: 10.1090/S0894-0347-2011-00727-3. |
[14] |
B. Dodson,
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 to appear in
Amer. J. Math. , arXiv: 1010.0040.
doi: 10.1090/S0894-0347-2011-00727-3. |
[15] |
B. Dodson,
Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Advances in Mathematics, 285 (2015), 1589-1618.
doi: 10.1016/j.aim.2015.04.030. |
[16] |
B. Dodson,
Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold,
arXiv: 1409. 1950. |
[17] |
J. Ginibre, T. Ozawa and G. Velo,
On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincare Phys. Theor., 60 (1994), 211-239.
|
[18] |
J. Ginibre and G. Velo,
Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl.(9), 64 (1985), 363-401.
|
[19] |
R. H. Goodman, R. E. Slusher and M. I. Weinstein,
Stopping light on a defect, J. Opt. Soc. Am. B, 19 (2002), 1635-1652.
doi: 10.1364/JOSAB.19.001635. |
[20] |
R. H. Goodman, M. I. Weinstein and P. J. Holmes,
Nonlinear propagation of light in one-dimensional periodic structures, J. Nonlinear Sci., 11 (2001), 123-168.
doi: 10.1007/s00332-001-0002-y. |
[21] |
Z. Hani and L. Thomann,
Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. Pure Appl. Math., 69 (2016), 1727-1776.
doi: 10.1002/cpa.21594. |
[22] |
K. Hepp,
The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277.
doi: 10.1007/BF01646348. |
[23] |
Y. Hong,
Scattering for a nonlinear Schrödinger equation with a potential, Comm. Pure Appl. Anal.(5), 15 (2016), 1571-1601.
doi: 10.3934/cpaa.2016003. |
[24] |
S. Ibrahim, N. Masmoudi and K. Nakanishi,
Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460.
doi: 10.2140/apde.2011.4.405. |
[25] |
J. L. Journe, A. Soffer and C. D. Sogge,
Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.
doi: 10.1002/cpa.3160440504. |
[26] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[27] |
S. Keraani,
On the defect of compactness for the Strichartz estimates for the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.
doi: 10.1006/jdeq.2000.3951. |
[28] |
C. E. 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. |
[29] |
R. Killip and M. Visan,
The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.
doi: 10.1353/ajm.0.0107. |
[30] |
R. Killip, M. Visan and X. Zhang,
The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Analysis and PDE, 1 (2009), 229-266.
doi: 10.2140/apde.2008.1.229. |
[31] |
D. Lafontaine,
Scattering for NLS with a potential on the line, Asymptotic Analysis, 100 (2016), 21-39.
doi: 10.3233/ASY-161384. |
[32] |
E. H. Lieb, R. Seiringer and J. Yngvason,
A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31.
doi: 10.1007/s002200100533. |
[33] |
H. P. McKean and J. Shatah,
The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math., 44 (1991), 1067-1080.
doi: 10.1002/cpa.3160440817. |
[34] |
K. Nakanishi,
Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, Journal of Functional Analysis, 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[35] |
F. Planchon and L. Vega,
Bilinear virial identities and applications, Ann. Sci. Ec. Norm. Super., 42 (2009), 261-290.
|
[36] |
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. |
[37] |
E. Ryckman and M. Visan,
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
doi: 10.1353/ajm.2007.0004. |
[38] |
W. Schlag,
Dispersive estimates for Schrödinger operators: A survey, Ann. of Math. Stud., 163 (2007), 255-285.
|
[39] |
A. Soffer and M. I. Weinstein,
Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.
doi: 10.1007/s002220050303. |
[40] |
H. Spohn,
Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.
doi: 10.1103/RevModPhys.52.569. |
[41] |
W. Strauss, Nonlinear scattering theory. Scattering theory in mathematical physics, Proceedings of the NATO Advanced Study Institue, (Denver, 1973), 53-78. NATO Advanced Science Institues, Volume C9. Reidel, Dordrecht, 1974. |
[42] |
W. Strauss,
Nonlinear scattering theory at low energy: Sequel, J. Funct. Anal., 43 (1981), 281-293.
doi: 10.1016/0022-1236(81)90019-7. |
[43] |
T. Tao,
Nonlinear Dispersive Equations: Local and Global Analysis
American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
[44] |
T. Tao, M. Visan and X. Zhang,
Minimal-mass blowup solutions of the mass-critical NLS, Forum Mathematicum, 20 (2008), 881-919.
doi: 10.1515/FORUM.2008.042. |
[45] |
T. Tao, M. Visan and X. Zhang,
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math J., 140 (2007), 165-202.
doi: 10.1215/S0012-7094-07-14015-8. |
[46] |
M. C. Vilela,
Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
[47] |
N. Visciglia,
On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.
doi: 10.4310/MRL.2009.v16.n5.a14. |
[48] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimemsions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
[1] |
Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286 |
[2] |
Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 |
[3] |
Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations and Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 |
[4] |
Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2837-2876. doi: 10.3934/dcdss.2020407 |
[5] |
Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure and Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 |
[6] |
Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215 |
[7] |
Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 |
[8] |
Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 |
[9] |
Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120 |
[10] |
Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure and Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 |
[11] |
Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 |
[12] |
Divyang G. Bhimani. The nonlinear Schrödinger equations with harmonic potential in modulation spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5923-5944. doi: 10.3934/dcds.2019259 |
[13] |
Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831 |
[14] |
Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253 |
[15] |
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 and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
[16] |
Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259 |
[17] |
Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7 |
[18] |
Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 |
[19] |
Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275 |
[20] |
Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]