Trudinger - Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi

Research output: Contribution to journalArticle

Abstract

The Trudinger-Moser inequality is a substitute for the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L<sup>∞</sup> . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails to hold on the whole plane, but a few modified versions are available. We prove a more precise version of the latter, giving necessary and sufficient conditions for boundedness, as well as for compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

Original languageEnglish (US)
Pages (from-to)819-835
Number of pages17
JournalJournal of the European Mathematical Society
Volume17
Issue number4
DOIs
StatePublished - 2015

Fingerprint

Trudinger-Moser Inequality
Growth Conditions
Nonlinear equations
Ground state
Sobolev Embedding
Nonlinear Klein-Gordon Equation
Best Constants
Substitute
Nonlinear Function
Ground State
Compactness
Boundedness
Nonlinear Equations
Exponent
Scaling
Decay
Necessary Conditions
Sufficient Conditions
Energy
Range of data

Keywords

  • Concentration compactness
  • Ground state
  • Nonlinear schrödinger equation
  • Sobolev critical exponent
  • Trudinger - Moser inequality

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Trudinger - Moser inequality on the whole plane with the exact growth condition. / Ibrahim, Slim; Masmoudi, Nader; Nakanishi, Kenji.

In: Journal of the European Mathematical Society, Vol. 17, No. 4, 2015, p. 819-835.

Research output: Contribution to journalArticle

@article{648d34db55a64a91a4025145b1628f6f,
title = "Trudinger - Moser inequality on the whole plane with the exact growth condition",
abstract = "The Trudinger-Moser inequality is a substitute for the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L∞ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails to hold on the whole plane, but a few modified versions are available. We prove a more precise version of the latter, giving necessary and sufficient conditions for boundedness, as well as for compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schr{\"o}dinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.",
keywords = "Concentration compactness, Ground state, Nonlinear schr{\"o}dinger equation, Sobolev critical exponent, Trudinger - Moser inequality",
author = "Slim Ibrahim and Nader Masmoudi and Kenji Nakanishi",
year = "2015",
doi = "10.4171/JEMS/519",
language = "English (US)",
volume = "17",
pages = "819--835",
journal = "Journal of the European Mathematical Society",
issn = "1435-9855",
publisher = "European Mathematical Society Publishing House",
number = "4",

}

TY - JOUR

T1 - Trudinger - Moser inequality on the whole plane with the exact growth condition

AU - Ibrahim, Slim

AU - Masmoudi, Nader

AU - Nakanishi, Kenji

PY - 2015

Y1 - 2015

N2 - The Trudinger-Moser inequality is a substitute for the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L∞ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails to hold on the whole plane, but a few modified versions are available. We prove a more precise version of the latter, giving necessary and sufficient conditions for boundedness, as well as for compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

AB - The Trudinger-Moser inequality is a substitute for the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L∞ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails to hold on the whole plane, but a few modified versions are available. We prove a more precise version of the latter, giving necessary and sufficient conditions for boundedness, as well as for compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

KW - Concentration compactness

KW - Ground state

KW - Nonlinear schrödinger equation

KW - Sobolev critical exponent

KW - Trudinger - Moser inequality

UR - http://www.scopus.com/inward/record.url?scp=84928253068&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928253068&partnerID=8YFLogxK

U2 - 10.4171/JEMS/519

DO - 10.4171/JEMS/519

M3 - Article

AN - SCOPUS:84928253068

VL - 17

SP - 819

EP - 835

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 4

ER -