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Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum
Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation
Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea |
For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [
References:
[1] |
J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia,
Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.
doi: 10.1007/s00220-017-2866-1. |
[2] |
J. Bellazzini, V. Georgiev and N. Visciglia,
Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.
doi: 10.1007/s00208-018-1666-z. |
[3] |
J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp.
doi: 10.1063/1.4726198. |
[4] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[6] |
T. Cazenave and P. L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[7] |
W. Choi, Y. Hong and J. Seok,
On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1241-1263.
doi: 10.1017/prm.2018.114. |
[8] |
R. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[9] |
J. Fröhlich, B. L. G. Jonsson and E. Lenzmann,
Effective dynamics for boson stars, Nonlinearity, 20 (2007), 1031-1075.
doi: 10.1088/0951-7715/20/5/001. |
[10] |
K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 473–478.
doi: 10.3934/proc.2015.0473. |
[11] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[12] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[13] |
E. Lenzmann,
Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.
doi: 10.1007/s11040-007-9020-9. |
[14] |
E. Lenzmann,
Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.
doi: 10.2140/apde.2009.2.1. |
[15] |
E. H. Lieb and H.-T. Yau,
The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.
doi: 10.1007/BF01217684. |
[16] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/s0294-1449(16)30422-x. |
[17] |
X. Luo and T. Yang, Stable solitary waves for pseudo-relativistic Hartree equations with short range potential, Nonlinear Anal., 207 (2021), Paper No. 112275, 13 pp.
doi: 10.1016/j.na.2021.112275. |
[18] |
J. Seok and Y. Hong,
Ground states to the generalized nonlinear schrödinger equations with bernstein symbols, Anal. Theory Appl., 37 (2021), 157-177.
doi: 10.4208/ata.2021.pr80.06. |
[19] |
Q. Shi and C. Peng,
Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133-144.
doi: 10.1016/j.na.2018.07.012. |
[20] |
M. I. Weinstein,
Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
show all references
References:
[1] |
J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia,
Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.
doi: 10.1007/s00220-017-2866-1. |
[2] |
J. Bellazzini, V. Georgiev and N. Visciglia,
Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension, Math. Ann., 371 (2018), 707-740.
doi: 10.1007/s00208-018-1666-z. |
[3] |
J. P. Borgna and D. F. Rial, Existence of ground states for a one-dimensional relativistic Schrödinger equation, J. Math. Phys., 53 (2012), 062301, 19 pp.
doi: 10.1063/1.4726198. |
[4] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[6] |
T. Cazenave and P. L. Lions,
Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[7] |
W. Choi, Y. Hong and J. Seok,
On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 1241-1263.
doi: 10.1017/prm.2018.114. |
[8] |
R. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[9] |
J. Fröhlich, B. L. G. Jonsson and E. Lenzmann,
Effective dynamics for boson stars, Nonlinearity, 20 (2007), 1031-1075.
doi: 10.1088/0951-7715/20/5/001. |
[10] |
K. Fujiwara, S. Machihara and T. Ozawa, Remark on a semirelativistic equation in the energy space, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 473–478.
doi: 10.3934/proc.2015.0473. |
[11] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[12] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[13] |
E. Lenzmann,
Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.
doi: 10.1007/s11040-007-9020-9. |
[14] |
E. Lenzmann,
Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.
doi: 10.2140/apde.2009.2.1. |
[15] |
E. H. Lieb and H.-T. Yau,
The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.
doi: 10.1007/BF01217684. |
[16] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/s0294-1449(16)30422-x. |
[17] |
X. Luo and T. Yang, Stable solitary waves for pseudo-relativistic Hartree equations with short range potential, Nonlinear Anal., 207 (2021), Paper No. 112275, 13 pp.
doi: 10.1016/j.na.2021.112275. |
[18] |
J. Seok and Y. Hong,
Ground states to the generalized nonlinear schrödinger equations with bernstein symbols, Anal. Theory Appl., 37 (2021), 157-177.
doi: 10.4208/ata.2021.pr80.06. |
[19] |
Q. Shi and C. Peng,
Wellposedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonlinear Anal., 178 (2019), 133-144.
doi: 10.1016/j.na.2018.07.012. |
[20] |
M. I. Weinstein,
Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
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