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.

中文翻译：

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受扰的 Moser-Trudinger 不等式何时承认极值？

在本文中，我们对主要在[17]中提出的几个问题感兴趣。我们考虑下面的扰动 Moser-Trudinger 不等式 $I\_\alpha^g(\Omega)$，在临界水平 $\alpha=4\pi$，其中 $g$，满足 $g(t)\to 0 $ as $t\to +\infty$，可以看作是相对于原始情况 $g\equiv 0$ 的扰动。在一些额外的假设下，基本上确保 $g$ 不会像 $t\to +\infty$ 那样振荡得太快，我们确定了 $g$ 上的一个新条件，使这种不等式具有极值。该条件涵盖了 [3,12,23] 中研究的情况 $g\equiv 0$。我们还证明了这个条件是尖锐的，如果它不满足，$I\_{4\pi}^g(\Omega)$ 可能没有极值。