### 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 language | English (US) |
---|---|

Pages (from-to) | 819-835 |

Number of pages | 17 |

Journal | Journal of the European Mathematical Society |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*17*(4), 819-835. https://doi.org/10.4171/JEMS/519

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

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 17, no. 4, pp. 819-835. https://doi.org/10.4171/JEMS/519

}

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 -