# Trudinger–Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim; Nader Masmoudi; Kenji Nakanishi

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 4, page 819-835
- ISSN: 1435-9855

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topIbrahim, Slim, Masmoudi, Nader, and Nakanishi, Kenji. "Trudinger–Moser inequality on the whole plane with the exact growth condition." Journal of the European Mathematical Society 017.4 (2015): 819-835. <http://eudml.org/doc/277652>.

@article{Ibrahim2015,

abstract = {Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\{\infty \}$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the 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.},

author = {Ibrahim, Slim, Masmoudi, Nader, Nakanishi, Kenji},

journal = {Journal of the European Mathematical Society},

keywords = {Sobolev critical exponent; Trudinger-Moser inequality; concentration compactness; nonlinear Schrödinger equation; ground state; Sobolev critical exponent; Trudinger-Moser inequality; concentration compactness; nonlinear Schrödinger equation; ground state},

language = {eng},

number = {4},

pages = {819-835},

publisher = {European Mathematical Society Publishing House},

title = {Trudinger–Moser inequality on the whole plane with the exact growth condition},

url = {http://eudml.org/doc/277652},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Ibrahim, Slim

AU - Masmoudi, Nader

AU - Nakanishi, Kenji

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

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 4

SP - 819

EP - 835

AB - Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty }$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the 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.

LA - eng

KW - Sobolev critical exponent; Trudinger-Moser inequality; concentration compactness; nonlinear Schrödinger equation; ground state; Sobolev critical exponent; Trudinger-Moser inequality; concentration compactness; nonlinear Schrödinger equation; ground state

UR - http://eudml.org/doc/277652

ER -

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