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Nonradial least energy solutions of the p-Laplace elliptic equations
Energy-critical NLS with potentials of quadratic growth
Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA |
$i\partial_t u = [-\tfrac{1}{2} Δ + V(x)] u ± |u|^{4/(d-2)} u, \ u(0)∈ Σ(\mathbf{R}^d),$ |
$Σ$ |
$\dot{H}^1 \cap |x|^{-1} L^2$ |
$V(x) = \tfrac{1}{2}|x|^2$ |
References:
[1] |
K. Asada and D. Fujiwara,
On some oscillatory integral transformations in $L^{2}(\textbf{R}^{n})$, Japan. J. Math. (N.S.), 4 (1978), 299-361.
|
[2] |
J. Bourgain,
Global wellposedness of defocusing critical nonlinear {S}chrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
doi: 10.1090/S0894-0347-99-00283-0. |
[3] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[4] |
R. Carles,
Nonlinear schrödinger equation with time-dependent potential, Commun. Math Sci., 9 (2011), 937-964.
doi: 10.4310/CMS.2011.v9.n4.a1. |
[5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[6] |
G. B. Folland, Harmonic Analysis in Phase Space vol. 122 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1989.
doi: 10.1515/9781400882427. |
[7] |
D. Fujiwara,
On the boundedness of integral transformations with highly oscillatory kernels, Proc. Japan Acad., 51 (1975), 96-99.
doi: 10.3792/pja/1195518693. |
[8] |
D. Fujiwara,
A construction of the fundamental solution for the Schrödinger equation, J. Analyse Math., 35 (1979), 41-96.
doi: 10.1007/BF02791062. |
[9] |
D. Fujiwara,
Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.
doi: 10.1215/S0012-7094-80-04734-1. |
[10] |
W. Hebisch,
A multiplier theorem for Schrödinger operators, Colloq. Math., 60/61 (1990), 659-664.
doi: 10.4064/cm-60-61-2-659-664. |
[11] |
A. D. Ionescu and B. Pausader,
The energy-critical defocusing NLS on $\mathbb{T}^3$, Duke Math. J., 161 (2012), 1581-1612.
doi: 10.1215/00127094-1593335. |
[12] |
A. D. Ionescu and B. Pausader,
Global well-posedness of the energy-critical defocusing {NLS} on $\mathbb{R}× \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.
doi: 10.1007/s00220-012-1474-3. |
[13] |
A. D. Ionescu, B. Pausader and G. Staffilani,
On the global well-posedness of energy-critical {S}chrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.
doi: 10.2140/apde.2012.5.705. |
[14] |
C. Jao,
The energy-critical quantum harmonic oscillator, Comm. Partial Differential Equations, 41 (2016), 79-133.
doi: 10.1080/03605302.2015.1095767. |
[15] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
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. |
[17] |
S. Keraani,
On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.
doi: 10.1016/j.jfa.2005.10.005. |
[18] |
R. Killip, S. Kwon, S. Shao and M. Visan,
On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.
doi: 10.3934/dcds.2012.32.191. |
[19] |
R. Killip, B. Stovall and M. Visan,
Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631.
doi: 10.1090/S0002-9947-2011-05536-4. |
[20] |
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. |
[21] |
R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer. Math. Soc., Providence, RI, 2013,325–437. |
[22] |
R. Killip, M. Visan and X. Zhang,
Quintic NLS in the exterior of a strictly convex obstacle, Amer. J. Math., 138 (2016), 1193-1346.
doi: 10.1353/ajm.2016.0039. |
[23] |
R. Killip, M. Visan and X. Zhang,
Energy-critical NLS with quadratic potentials, Comm. Partial Differential Equations, 34 (2009), 1531-1565.
doi: 10.1080/03605300903328109. |
[24] |
Y.-G. Oh,
Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.
doi: 10.1016/0022-0396(89)90123-X. |
[25] |
E. Ryckman and M. Visan,
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\Bbb R^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
doi: 10.1353/ajm.2007.0004. |
[26] |
M. E. Taylor, Tools for PDE vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Pseudodifferential operators, paradifferential operators, and layer potentials. |
[27] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
[28] |
J. Zhang,
Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.
doi: 10.1023/A:1026437923987. |
show all references
References:
[1] |
K. Asada and D. Fujiwara,
On some oscillatory integral transformations in $L^{2}(\textbf{R}^{n})$, Japan. J. Math. (N.S.), 4 (1978), 299-361.
|
[2] |
J. Bourgain,
Global wellposedness of defocusing critical nonlinear {S}chrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
doi: 10.1090/S0894-0347-99-00283-0. |
[3] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[4] |
R. Carles,
Nonlinear schrödinger equation with time-dependent potential, Commun. Math Sci., 9 (2011), 937-964.
doi: 10.4310/CMS.2011.v9.n4.a1. |
[5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[6] |
G. B. Folland, Harmonic Analysis in Phase Space vol. 122 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1989.
doi: 10.1515/9781400882427. |
[7] |
D. Fujiwara,
On the boundedness of integral transformations with highly oscillatory kernels, Proc. Japan Acad., 51 (1975), 96-99.
doi: 10.3792/pja/1195518693. |
[8] |
D. Fujiwara,
A construction of the fundamental solution for the Schrödinger equation, J. Analyse Math., 35 (1979), 41-96.
doi: 10.1007/BF02791062. |
[9] |
D. Fujiwara,
Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980), 559-600.
doi: 10.1215/S0012-7094-80-04734-1. |
[10] |
W. Hebisch,
A multiplier theorem for Schrödinger operators, Colloq. Math., 60/61 (1990), 659-664.
doi: 10.4064/cm-60-61-2-659-664. |
[11] |
A. D. Ionescu and B. Pausader,
The energy-critical defocusing NLS on $\mathbb{T}^3$, Duke Math. J., 161 (2012), 1581-1612.
doi: 10.1215/00127094-1593335. |
[12] |
A. D. Ionescu and B. Pausader,
Global well-posedness of the energy-critical defocusing {NLS} on $\mathbb{R}× \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.
doi: 10.1007/s00220-012-1474-3. |
[13] |
A. D. Ionescu, B. Pausader and G. Staffilani,
On the global well-posedness of energy-critical {S}chrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.
doi: 10.2140/apde.2012.5.705. |
[14] |
C. Jao,
The energy-critical quantum harmonic oscillator, Comm. Partial Differential Equations, 41 (2016), 79-133.
doi: 10.1080/03605302.2015.1095767. |
[15] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[16] |
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. |
[17] |
S. Keraani,
On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.
doi: 10.1016/j.jfa.2005.10.005. |
[18] |
R. Killip, S. Kwon, S. Shao and M. Visan,
On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., 32 (2012), 191-221.
doi: 10.3934/dcds.2012.32.191. |
[19] |
R. Killip, B. Stovall and M. Visan,
Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631.
doi: 10.1090/S0002-9947-2011-05536-4. |
[20] |
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. |
[21] |
R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer. Math. Soc., Providence, RI, 2013,325–437. |
[22] |
R. Killip, M. Visan and X. Zhang,
Quintic NLS in the exterior of a strictly convex obstacle, Amer. J. Math., 138 (2016), 1193-1346.
doi: 10.1353/ajm.2016.0039. |
[23] |
R. Killip, M. Visan and X. Zhang,
Energy-critical NLS with quadratic potentials, Comm. Partial Differential Equations, 34 (2009), 1531-1565.
doi: 10.1080/03605300903328109. |
[24] |
Y.-G. Oh,
Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274.
doi: 10.1016/0022-0396(89)90123-X. |
[25] |
E. Ryckman and M. Visan,
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\Bbb R^{1+4}$, Amer. J. Math., 129 (2007), 1-60.
doi: 10.1353/ajm.2007.0004. |
[26] |
M. E. Taylor, Tools for PDE vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Pseudodifferential operators, paradifferential operators, and layer potentials. |
[27] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
[28] |
J. Zhang,
Stability of attractive Bose-Einstein condensates, J. Statist. Phys., 101 (2000), 731-746.
doi: 10.1023/A:1026437923987. |
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