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Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencies
About this Title
Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 272, Number 1331
ISBNs: 978-1-4704-4872-1 (print); 978-1-4704-6747-0 (online)
DOI: https://doi.org/10.1090/memo/1331
Published electronically: September 21, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Properties of Ground State
- 3. Proof of Theorem
- 4. Decomposition around ground state
- 5. Ejection lemma
- 6. Modified distance function
- 7. One-pass theorem
- 8. Proof of Theorem
- A. Existence of ground state
- B. Fundamental properties of the linearized operators
- C. Inequalities for the radial functions
- D. Small-data theory
- E. Long-time perturbation theory
- F. Table of notation
Abstract
We consider the combined power-type nonlinear Schrödinger equations with energy-critical growth, and study the solutions slightly above the ground state threshold at low frequencies, so that we obtain a so-called nine-set theory developed by Nakanishi and Schlag.- Takafumi Akahori and Hayato Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math. 53 (2013), no. 3, 629–672. MR 3102564, DOI 10.1215/21562261-2265914
- Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi, and Hayato Nawa, Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth, Differential Integral Equations 25 (2012), no. 3-4, 383–402. MR 2917888
- Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi, and Hayato Nawa, Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth, Selecta Math. (N.S.) 19 (2013), no. 2, 545–609. MR 3090237, DOI 10.1007/s00029-012-0103-5
- 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
- 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, DOI 10.1007/BF00250555
- Haïm Brezis and Elliott H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys. 96 (1984), no. 1, 97–113. MR 765961
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
- Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
- Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal. 39 (2007/08), no. 4, 1070–1111. MR 2368894, DOI 10.1137/050648389
- 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, DOI 10.1016/j.jde.2016.04.031
- Thomas Duyckaerts, Justin Holmer, and Svetlana Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett. 15 (2008), no. 6, 1233–1250. MR 2470397, DOI 10.4310/MRL.2008.v15.n6.a13
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^{n}$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
- Manoussos Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math. 41 (1988), no. 6, 747–774. MR 948770, DOI 10.1002/cpa.3160410602
- Justin Holmer and Svetlana Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), no. 2, 435–467. MR 2421484, DOI 10.1007/s00220-008-0529-y
- Slim Ibrahim, Nader Masmoudi, and Kenji Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE 4 (2011), no. 3, 405–460. MR 2872122, DOI 10.2140/apde.2011.4.405
- 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
- Yoshitsugu Kabeya and Kazunaga Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $\mathbf R^N$ and Séré’s non-degeneracy condition, Comm. Partial Differential Equations 24 (1999), no. 3-4, 563–598. MR 1683050, DOI 10.1080/03605309908821434
- 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, DOI 10.1007/s00222-006-0011-4
- Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), no. 2, 353–392. MR 1855973, DOI 10.1006/jdeq.2000.3951
- Rowan Killip, Tadahiro Oh, Oana Pocovnicu, and Monica Vişan, Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on $\Bbb {R}^3$, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 469–548. MR 3634031, DOI 10.1007/s00205-017-1109-0
- 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, DOI 10.1353/ajm.0.0107
- Rowan Killip and Monica Vişan, Nonlinear Schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437. MR 3098643, DOI 10.1007/s00208-013-0960-z
- Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
- K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), no. 5, 2299–2333. MR 2756065, DOI 10.1016/j.jde.2010.10.027
- K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 1–45. MR 2898769, DOI 10.1007/s00526-011-0424-9
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Jalal Shatah and Walter Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), no. 2, 173–190. MR 804458
- Terence Tao and Monica Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations (2005), No. 118, 28. MR 2174550
- 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, DOI 10.1080/03605300701588805
- Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
- Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, DOI 10.1137/0516034
- Jian Zhang and WenMing Zou, The critical case for a Berestycki-Lions theorem, Sci. China Math. 57 (2014), no. 3, 541–554. MR 3166237, DOI 10.1007/s11425-013-4687-9