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

TL;DR

This paper investigates the existence of extremals for a perturbed Moser-Trudinger inequality at the critical level, identifying a sharp condition on the perturbation function g that determines when extremals exist.

Contribution

It introduces a new condition on the perturbation g ensuring the existence of extremals for the perturbed inequality, extending previous results and covering the unperturbed case.

Findings

A new criterion on g for extremal existence is established.

The criterion is shown to be sharp, with failure leading to no extremals.

The results generalize previous work on the unperturbed case.

Abstract

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.