Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth

被引：1
作者：
Takafumi Akahori
Slim Ibrahim
Hiroaki Kikuchi
Hayato Nawa
机构：
[1] Shizuoka University,Faculty of Engineering
[2] University of Victoria,Department of Mathematics and Statistics
[3] Tsuda College,Department of Mathematics
[4] Osaka University,Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science
来源：
Selecta Mathematica | 2013年 / 19卷
关键词：
Nonlinear Schrödinger equation; Scattering problem; Profile decomposition; Virial identity; Variational methods; Ground state; 35B40; 35B60; 35Q55;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We study the energy-critical focusing nonlinear Schrödinger equation with an energy-subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig and Merle (Invent Math 166:645–675, 2006).
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页码：545 / 609
页数：64
相关论文
共 36 条
[1]  
Akahori T(2012)Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth Differ. Integral Equ. 25 383-402
[2]  
Ibrahim S(1976)Problémes isopérimétriques et espaces de Sobolev J. Differ. Geom. 11 573-598
[3]  
Kikuchi H(1981)Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires C. R. Acad. Sci. Paris Sér I Math. 293 489-492
[4]  
Nawa H(1983)A relation between pointwise convergence of functions and convergence of functionals Proc. Am. Math. Soc. 88 485-489
[5]  
Aubin T(1983)Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Commun. Pure Appl. Math. 36 437-477
[6]  
Berestycki H(1990)The Cauchy problem for the critical nonlinear Schödinger equation in Nonlinear Anal. 14 807-836
[7]  
Cazenave T(1991)Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation J. Funct. Anal. 100 87-109
[8]  
Brézis H.(1992)Homoclinic type solutions for a semilinear elliptic PDE on Commun. Pure Appl. Math. 45 1217-1269
[9]  
Lieb E.H.(2008)Scattering for the non-radial 3D cubic nonlinear Schrödinger equation Math. Res. Lett. 15 1233-1250
[10]  
Brezis H(2008)A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation Commun. Math. Phys. 282 435-467
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