Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities
Authors:
Noriyoshi Fukaya and Masayuki Hayashi
Journal:
Trans. Amer. Math. Soc. 374 (2021), 1421-1447
MSC (2010):
Primary 35Q55; Secondary 35A15
DOI:
https://doi.org/10.1090/tran/8269
Published electronically:
November 25, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We consider a nonlinear Schrödinger equation with double power nonlinearity
![]() |
where




- [1] Jacopo Bellazzini, Rupert L. Frank, and Nicola Visciglia, Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems, Math. Ann. 360 (2014), no. 3-4, 653–673. MR 3273640, https://doi.org/10.1007/s00208-014-1046-2
- [2] Henri Berestycki and Thierry Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492 (French, with English summary). MR 646873
- [3] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, https://doi.org/10.1007/BF00250555
- [4] Haïm Brézis and Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. MR 699419, https://doi.org/10.1090/S0002-9939-1983-0699419-3
- [5] Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047
- [6] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. MR 677997
- [7] Xing Cheng, Changxing Miao, and Lifeng Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations 261 (2016), no. 6, 2881–2934. MR 3527618, https://doi.org/10.1016/j.jde.2016.04.031
- [8] Mathieu Colin and Masahito Ohta, Stability of solitary waves for derivative nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 5, 753–764 (English, with English and French summaries). MR 2259615, https://doi.org/10.1016/j.anihpc.2005.09.003
- [9] Mathieu Colin and Masahito Ohta, Instability of ground states for a quasilinear Schrödinger equation, Differential Integral Equations 27 (2014), no. 7-8, 613–624. MR 3200755
- [10] E. N. Dancer and Sanjiban Santra, Singular perturbed problems in the zero mass case: asymptotic behavior of spikes, Ann. Mat. Pura Appl. (4) 189 (2010), no. 2, 185–225. MR 2602148, https://doi.org/10.1007/s10231-009-0105-x
- [11] E. N. Dancer, Sanjiban Santra, and Juncheng Wei, Asymptotic behavior of the least energy solution of a problem with competing powers, J. Funct. Anal. 261 (2011), no. 8, 2094–2134. MR 2824573, https://doi.org/10.1016/j.jfa.2011.06.005
- [12] Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875 (English, with English and French summaries). MR 1940370, https://doi.org/10.1016/S0021-7824(02)01266-7
- [13] Thomas Duyckaerts, Hao Jia, Carlos Kenig, and Frank Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal. 27 (2017), no. 4, 798–862. MR 3678502, https://doi.org/10.1007/s00039-017-0418-7
- [14] Noriyoshi Fukaya, Masayuki Hayashi, and Takahisa Inui, A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation, Anal. PDE 10 (2017), no. 5, 1149–1167. MR 3668587, https://doi.org/10.2140/apde.2017.10.1149
- [15] Reika Fukuizumi, Remarks on the stable standing waves for nonlinear Schrödinger equations with double power nonlinearity, Adv. Math. Sci. Appl. 13 (2003), no. 2, 549–564. MR 2029931
- [16] Reika Fukuizumi and Masahito Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations 16 (2003), no. 1, 111–128. MR 1948875
- [17] Reika Fukuizumi and Masahito Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations 16 (2003), no. 6, 691–706. MR 1973275
- [18] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
- [19] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, https://doi.org/10.1016/0022-1236(87)90044-9
- [20] Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), no. 2, 308–348. MR 1081647, https://doi.org/10.1016/0022-1236(90)90016-E
- [21] Masayuki Hayashi, Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, 1331–1360. MR 3985546, https://doi.org/10.1016/j.anihpc.2018.12.003
- [22] M. Hayashi, Potential well theory for the derivative nonlinear Schrödinger equation, to appear in Anal. PDE. preprint (2019).
- [23] M. Hayashi, Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach, preprint (2019).
- [24] Iliya D. Iliev and Kiril P. Kirchev, Stability and instability of solitary waves for one-dimensional singular Schrödinger equations, Differential Integral Equations 6 (1993), no. 3, 685–703. MR 1202566
- [25] Carlos E. Kenig and Frank 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), no. 3, 645–675. MR 2257393, https://doi.org/10.1007/s00222-006-0011-4
- [26] Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), no. 2, 361–424. MR 2654778, https://doi.org/10.1353/ajm.0.0107
- [27] Stefan Le Coz, A note on Berestycki-Cazenave’s classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud. 8 (2008), no. 3, 455–463. MR 2426909, https://doi.org/10.1515/ans-2008-0302
- [28] Yi Li and Wei-Ming Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ, Comm. Partial Differential Equations 18 (1993), no. 5-6, 1043–1054. MR 1218528, https://doi.org/10.1080/03605309308820960
- [29] Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, https://doi.org/10.2307/2007032
- [30] Elliott H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), no. 3, 441–448. MR 724014, https://doi.org/10.1007/BF01394245
- [31] Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225
- [32] Masaya Maeda, Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity, Kodai Math. J. 31 (2008), no. 2, 263–271. MR 2435895, https://doi.org/10.2996/kmj/1214442798
- [33] Masahito Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity, Kodai Math. J. 18 (1995), no. 1, 68–74. MR 1317007, https://doi.org/10.2996/kmj/1138043354
- [34] Masahito Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor. 62 (1995), no. 1, 69–80 (English, with English and French summaries). MR 1313361
- [35] Masahito Ohta and Takahiro Yamaguchi, Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity, SUT J. Math. 51 (2015), no. 1, 49–58. MR 3409057
- [36] Patrizia Pucci and James Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (1998), no. 2, 501–528. MR 1647924, https://doi.org/10.1512/iumj.1998.47.1517
- [37] James Serrin and Moxun Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), no. 3, 897–923. MR 1803216, https://doi.org/10.1512/iumj.2000.49.1893
- [38] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
- [39] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, https://doi.org/10.1007/BF02418013
- [40] Terence Tao, Monica Visan, and Xiaoyi Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1281–1343. MR 2354495, https://doi.org/10.1080/03605300701588805
- [41] Laurent Véron, Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans 𝑅^{𝑁}, Ann. Mat. Pura Appl. (4) 127 (1981), 25–50 (French, with English summary). MR 633393, https://doi.org/10.1007/BF01811717
- [42] Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
- [43] Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, https://doi.org/10.1137/0516034
- [44] Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. MR 820338, https://doi.org/10.1002/cpa.3160390103
- [45] Yifei Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. PDE 8 (2015), no. 5, 1101–1112. MR 3393674, https://doi.org/10.2140/apde.2015.8.1101
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35Q55, 35A15
Retrieve articles in all journals with MSC (2010): 35Q55, 35A15
Additional Information
Noriyoshi Fukaya
Affiliation:
Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan
Email:
fukaya@rs.tus.ac.jp
Masayuki Hayashi
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email:
hayashi@kurims.kyoto-u.ac.jp
DOI:
https://doi.org/10.1090/tran/8269
Keywords:
Nonlinear Schr\"odinger equation,
standing waves,
orbital instability,
variational methods
Received by editor(s):
February 11, 2020
Received by editor(s) in revised form:
July 3, 2020
Published electronically:
November 25, 2020
Additional Notes:
The first author was supported by JSPS KAKENHI Grant Number 20K14349.
The second author was supported by JSPS KAKENHI Grant Number JP19J01504.
Article copyright:
© Copyright 2020
American Mathematical Society