November  2016, 15(6): 2023-2058. doi: 10.3934/cpaa.2016026

Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

1. 

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University, Suita, Osaka 565-0871, Japan

2. 

Graduate School of Mathematics, Nagoya University, Japan

Received  October 2015 Revised  June 2016 Published  September 2016

Consider the focusing energy critical Schrödinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
Citation: Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
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show all references

References:
[1]

J. Math. Pures. Appl., 55 (1976), 269-296.  Google Scholar

[2]

Amer. J. Math., 121 (1999), 131-175.  Google Scholar

[3]

J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar

[4]

Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[6]

Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.  Google Scholar

[7]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.   Google Scholar

[8]

Comm. Pure. Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[9]

Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[10]

J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar

[11]

Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

Amer. J. Math., 135 (2013), 935-965. doi: 10.1353/ajm.2013.0034.  Google Scholar

[13]

Discrete, Cont, Dyn. Syst., 33 (2013), 2423-2450.  Google Scholar

[14]

Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[15]

Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.  Google Scholar

[16]

J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.  Google Scholar

[17]

J. Diff. Eq., 250 (2011), 2299-2333. doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[18]

Arch. Rational Mech. Analysis, 203 (2012), 809-851. doi: 10.1007/s00205-011-0462-7.  Google Scholar

[19]

Calc. Var. and PDE, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar

[20]

Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.  Google Scholar

[21]

J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[22]

Discrete, Cont, Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar

[23]

Ann. Mat. Pura. Appl., 110 (1976), 353-372.  Google Scholar

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