We consider the stability of standing waves for the focusing nonlinear Schrödinger equation with an inverse-square potential. Using the profile decomposition arguments, we show that in the L2-subcritical case, i.e., , the sets of ground state standing waves are orbitally stable. In the L2-critical case, i.e., , we show that ground state standing waves are strongly unstable by blow-up.
REFERENCES
1.
H.
Berestycki
and T.
Cazenave
, “Instabilité des états stationaires dans les equations de Schrödinger equations et de Klein-Gordon non linéaires
,” C. R. Acad. Sci. Paris
293
, 489
–492
(1981
).2.
A.
Bensouilah
, “L2 concentration of blow-up solutions for the mass-critical NLS with inverse-square potential
,” preprint arXiv:1803.05944 (2018
).3.
A.
Bensouilah
and V. D.
Dinh
, “Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schrödinger equation with inverse-square potential
,” preprint arXiv:1804.08752 (2018
).4.
T.
Cazenave
and P. L.
Lions
, “Orbital stability of standing waves for some nonlinear Schrdinger equations
,” Commun. Math. Phys.
85
, 549
–561
(1982
).5.
J. M.
Bouclet
and H.
Mizutani
, “Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities
,” Trans. Am. Math. Soc.
370
, 7293
–7333
(2018
).6.
N.
Burq
, F.
Planchon
, J.
Stalker
, and A. S.
Tahvildar-Zadeh
, “Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential
,” J. Funct. Anal.
203
, 519
–549
(2003
).7.
H.
Brézis
and E. H.
Lieb
, “A relation between pointwise convergence of functions and convergence of functionals
,” Proc. Am. Math. Soc.
88
, 486
–490
(1983
).8.
H. E.
Camblong
, L. N.
Epele
, H.
Fanchiotti
, and C. A.
Garcia Canal
, “Quantum anomaly in molecular physics
,” Phys. Rev. Lett.
87
, 220402
(2001
).9.
K. M.
Case
, “Singular potentials
,” Phys. Rev.
80
, 797
–806
(1950
).10.
T.
Cazenave
, Semilinear Schrödinger Equations
, Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, (American Mathematical Society
, 2003
), Vol. 10.11.
E.
Csobo
and F.
Genoud
, “Minimal mass blow-up solutions for the L2 critical NLS with inverse-square potential
,” Nonlinear Anal.
168
, 110
–129
(2018
).12.
V. D.
Dinh
, “Global existence and blow-up for a class of the focusing nonlinear Schrödinger equation with inverse-square potential
,” preprint arXiv:1711.04792 (2017
).13.
B.
Feng
and H.
Zhang
, “Stability of standing waves for the fractional Schrödinger-Hartree equation
,” J. Math. Anal. Appl.
460
, 352
–364
(2018
).14.
P.
Gérard
, “Description du defaut de compacité de l’injection de Sobolev
,” ESAIM Control Optim. Calc. Var.
3
, 213
–233
(1998
).15.
R. T.
Glassey
, “On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation
,” J. Math. Phys.
18
, 1794
–1797
(1977
).16.
Q.
Guo
and S. H.
Zhu
, “Sharp threshold of blow-up and scattering for the fractional Hartree equation
,” J. Differ. Equations
264
, 2802
–2832
(2018
).17.
T.
Hmidi
and S.
Keraani
, “Blow-up theory for the critical nonlinear Schrödinger equation revisited
,” Int. Math. Res. Not.
2005
(46
), 2815
–2828
.18.
H.
Kalf
, U. W.
Schmincke
, J.
Walter
and R.
Wust
, “On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials
,” in Spectral Theory and Differential Equations
, Lecture Notes in Mathematics (Springer
, Berlin
, 1975
), Vol. 448, pp. 182
–226
.19.
R.
Killip
, J.
Murphy
, M.
Visan
, and J.
Zheng
, “The focusing cubic NLS with inverse-square potential in three space dimensions
,” Differ. Integr. Equations
30
, 161
–206
(2017
).20.
R.
Killip
, C.
Miao
, M.
Visan
, J.
Zhang
, and J.
Zheng
, “Sobolev spaces adapted to the Schrödinger operator with inverse-square potential
,” Math. Z.
288
, 1273
–1298
(2017
).21.
R.
Killip
, C.
Miao
, M.
Visan
, J.
Zhang
, and J.
Zheng
, “The energy-critical NLS with inverse-square potential
,” Discrete Contin. Dyn. Syst.
37
, 3831
–3866
(2017
).22.
J.
Lu
, C.
Miao
, and J.
Murphy
, “Scattering in H1 for the intercritical NLS with an inverse-square potential
,” J. Differ. Equations
264
(5
), 3174
–3211
(2018
).23.
T.
Ogawa
and Y.
Tsutsumi
, “Blow-up of H1-solutions for the nonlinear Schrödinger equation
,” J. Differ. Equations
92
, 317
–330
(1991
).24.
N.
Okazawa
, T.
Suzuki
, and T.
Yokota
, “Energy methods for abstract nonlinear Schrödinger equations
,” Evol. Equations Control Theory
1
, 337
–354
(2012
).25.
C.
Peng
and Q.
Shi
, “Stability of standing waves for the fractional nonlinear Schrödinger equation
,” J. Math. Phys.
59
, 011508
(2018
).26.
P.
Trachanas
and N. B.
Zographopoulos
, “Orbital stability for the Schrödinger operator involving inverse square potential
,” J. Differ. Equations
259
, 4989
–5016
(2015
).27.
M.
Weinstein
, “Nonlinear Schrödinger equations and sharp interpolation estimates
,” Commun. Math. Phys.
87
, 567
–576
(1983
).28.
J.
Zhang
and J.
Zheng
, “Scattering theory for nonlinear Schrödinger with inverse-square potential
,” J. Funct. Anal.
267
, 2907
–2932
(2014
).29.
J.
Zhang
, “Stability of attractive Bose-Einstein condensates
,” J. Stat. Phys.
101
, 731
–746
(2000
).30.
G.
Fibich
, The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse
(Springer
, 2015
).31.
S. H.
Zhu
, J.
Zhang
, and H.
Yang
, “Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation
,” Dyn. Partial Differ. Equations
7
(2
), 187
–205
(2010
).32.
S. H.
Zhu
, “On the blow-up solutions for the nonlinear fractional Schrödinger equation
,” J. Differ. Equations
261
, 1506
–1531
(2016
).33.
S. H.
Zhu
, “Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities
,” J. Evol. Equations
17
, 1003
–1021
(2017
).© 2018 Author(s).
2018
Author(s)
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