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Article Dans Une Revue Analysis & PDE Année : 2020

When does a perturbed Moser-Trudinger inequality admit an extremal?

Résumé

In this paper, we are interested in several questions raised mainly in [17]. We consider the perturbed Moser-Trudinger inequality $I_\alpha^g(\Omega)$ below, at the critical level $\alpha=4\pi$, where $g$, satisfying $g(t)\to 0$ as $t\to +\infty$, can be seen as a perturbation with respect to the original case $g\equiv 0$. Under some additional assumptions, ensuring basically that $g$ does not oscillates too fast as $t\to +\infty$, we identify a new condition on $g$ for this inequality to have an extremal. This condition covers the case $g\equiv 0$ studied in [3,12,23]. We prove also that this condition is sharp in the sense that, if it is not satisfied, $I_{4\pi}^g(\Omega)$ may have no extremal.
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Dates et versions

hal-01701744 , version 1 (06-02-2018)
hal-01701744 , version 2 (13-05-2019)

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Citer

Pierre-Damien Thizy. When does a perturbed Moser-Trudinger inequality admit an extremal?. Analysis & PDE, 2020, ⟨10.2140/apde.2020.13.1371⟩. ⟨hal-01701744v2⟩
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