doi: 10.3934/dcdsb.2022125
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Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Leijin Cao

Received  August 2021 Revised  May 2022 Early access July 2022

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: Leijin Cao. Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022125
References:
[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. CamblongL. N. EpeleH. 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. GrillakisJ. 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. GrillakisJ. 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. KillipC. MiaoM. VisanJ. 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. KillipC. MiaoM. VisanJ. 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. KillipJ. MurphyM. 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. LuC. 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. OkazawaT. 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. TaoM. 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.

show all references

References:
[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. CamblongL. N. EpeleH. 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. GrillakisJ. 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. GrillakisJ. 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. KillipC. MiaoM. VisanJ. 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. KillipC. MiaoM. VisanJ. 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. KillipJ. MurphyM. 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. LuC. 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. OkazawaT. 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. TaoM. 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.

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