In this paper, we consider the existence of stable standing waves for the nonlinear Schrödinger equation with combined power nonlinearities and the Hardy potential. In the $ L^2 $-critical case, we show that the set of energy minimizers is orbitally stable by using concentration compactness principle. In the $ L^2 $-supercritical case, we show that all energy minimizers correspond to local minima of the associated energy functional and we prove that the set of energy minimizers is orbitally stable.
Citation: |
[1] |
A. Bensouilah, $L^2$ concentration of blow-up solutions for the mass-critical NLS with inverse-square potential, Bull. Belg. Math. Soc. Simon Stevin, 26 (2019), 759-771.
doi: 10.36045/bbms/1579402821.![]() ![]() ![]() |
[2] |
A. Bensouilah, V. D. Dinh and S. Zhu, On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential, J. Math. Phys., 59 (2018), 101505, 18 pp.
doi: 10.1063/1.5038041.![]() ![]() ![]() |
[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: 10.1090/S0002-9939-1983-0699419-3.![]() ![]() ![]() |
[4] |
H. E. Camblong, L. N. Epele, H. Fanchiotti and C. A. Garcia Canal, Quantum anomaly in molecular physics, Phys. Rev. Lett., 87 (2001), 220402.
![]() |
[5] |
K. M. Case, Singular potentials, Phys. Rev., 80 (1950), 797-806.
doi: 10.1103/PhysRev.80.797.![]() ![]() ![]() |
[6] |
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504.![]() ![]() ![]() |
[7] |
E. Csobo and F. Genoud, Minimal mass blow-up solutions for the $L^2$ critical NLS with inverse-square potential, Nonlinear Anal., 168 (2018), 110-129.
doi: 10.1016/j.na.2017.11.008.![]() ![]() ![]() |
[8] |
V. D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl., 468 (2018), 270-303.
doi: 10.1016/j.jmaa.2018.08.006.![]() ![]() ![]() |
[9] |
V. D. Dinh, On instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential, Complex Var. Elliptic Equ., 66 (2021), 1699-1716.
doi: 10.1080/17476933.2020.1779235.![]() ![]() ![]() |
[10] |
V. D. Dinh, On nonlinear Schrödinger equations with attractive inverse-power potentials, Topol. Methods Nonlinear Anal., 57 (2021), 489-523.
doi: 10.12775/tmna.2020.046.![]() ![]() ![]() |
[11] |
B. Feng, L. Cao and J. Liu, Existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation, Appl. Math. Lett., 115 (2021), 106952, 7 pp.
doi: 10.1016/j.aml.2020.106952.![]() ![]() ![]() |
[12] |
B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.
doi: 10.1016/j.camwa.2017.12.025.![]() ![]() ![]() |
[13] |
B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364.
doi: 10.1016/j.jmaa.2017.11.060.![]() ![]() ![]() |
[14] |
R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 111-128.
![]() ![]() |
[15] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9.![]() ![]() ![]() |
[16] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.
doi: 10.1016/0022-1236(90)90016-E.![]() ![]() ![]() |
[17] |
L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliplic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1.![]() ![]() ![]() |
[18] |
L. Jeanjean, J. Jendrej, T. T. Le and N. Visciglia, Orbital stability of ground states for a Sobolev critical Schrödinger equation, preprint, 2020. arXiv: 2008.12084.
![]() |
[19] |
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 (2018), 1273-1298.
doi: 10.1007/s00209-017-1934-8.![]() ![]() ![]() |
[20] |
R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831-3866.
doi: 10.3934/dcds.2017162.![]() ![]() ![]() |
[21] |
R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inverse-square potential in three space dimensions, Differential Integral Equations, 30 (2017), 161-206.
![]() ![]() |
[22] |
M. Lewin and S. R. Nodari, The double-power nonlinear Schrödinger equation and its generalizations: Uniqueness, non-degeneracy and applications, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 197, 49 pp.
doi: 10.1007/s00526-020-01863-w.![]() ![]() ![]() |
[23] |
X. Li and J. Zhao, Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl., 79 (2020), 303-316.
doi: 10.1016/j.camwa.2019.06.030.![]() ![]() ![]() |
[24] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/s0294-1449(16)30428-0.![]() ![]() ![]() |
[25] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/s0294-1449(16)30422-x.![]() ![]() ![]() |
[26] |
J. Lu, C. Miao and J. Murphy, Scattering in $H^1$ for the intercritical NLS with an inverse-square potential, J. Differential Equations, 264 (2018), 3174-3211.
doi: 10.1016/j.jde.2017.11.015.![]() ![]() ![]() |
[27] |
V. Moncrief, Odd-parity stability of a Reissner-Nordström black hole, Phys. Rev. D, 9 (1974), 2707-2709.
![]() |
[28] |
D. Mukherjee, P. T. Nam and P.-T. Nguyen, Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential, J. Funct. Anal., 281 (2021), 109092, 45 pp.
doi: 10.1016/j.jfa.2021.109092.![]() ![]() ![]() |
[29] |
M. Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74.
doi: 10.2996/kmj/1138043354.![]() ![]() ![]() |
[30] |
N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory, 1 (2012), 337-354.
doi: 10.3934/eect.2012.1.337.![]() ![]() ![]() |
[31] |
N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), 6941-6987.
doi: 10.1016/j.jde.2020.05.016.![]() ![]() ![]() |
[32] |
N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal., 279 (2020), 108610, 43 pp.
doi: 10.1016/j.jfa.2020.108610.![]() ![]() ![]() |
[33] |
A. Stefanov, On the normalized ground states of second order PDE's with mixed power non-linearities, Commun. Math. Phys., 369 (2019), 929-971.
doi: 10.1007/s00220-019-03484-7.![]() ![]() ![]() |
[34] |
T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805.![]() ![]() ![]() |
[35] |
G. P. Trachanas and N. B. Zographopoulos, Orbital stability for the Schrödinger operator involving inverse square potential, J. Differential Equations, 259 (2015), 4989-5016.
doi: 10.1016/j.jde.2015.06.013.![]() ![]() ![]() |
[36] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation eatimates, Commun. Math. Phys., 87 (1983), 567-576.
![]() ![]() |
[37] |
F. J. Zerilli, Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D, 2 (1970), 2141-2160.
doi: 10.1103/PhysRevD.2.2141.![]() ![]() ![]() |
[38] |
F. J. Zerilli, Perturbation analysis for gravitational and electromagnetic radiation in a Reissner-Nordström geometry, Phys. Rev. D, 9 (1974), 860-868.
![]() |
[39] |
J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger equations with inverse-square potential, J. Funct. Anal., 267 (2014), 2907-2932.
doi: 10.1016/j.jfa.2014.08.012.![]() ![]() ![]() |
[40] |
J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030.
doi: 10.1007/s10884-015-9477-3.![]() ![]() ![]() |
[41] |
J. Zheng, Focusing NLS with inverse square potential, J. Math. Phys., 59 (2018), 111502, 14 pp.
doi: 10.1063/1.5054167.![]() ![]() ![]() |
[42] |
S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021.
doi: 10.1007/s00028-016-0363-1.![]() ![]() ![]() |