Moser–Trudinger inequality in grand Lebesgue space

Abstract

Let $n\in \mathbb{N}$, $n\ge 2$ and let $\Omega \subset {\mathbb{R}}^n$ be a bounded domain. We study sharp constants for the Moser–Trudinger inequality in the Sobolev-type space $W_0{L}^{n)}(\Omega )$, where $L^{n)}\Omega$ is so called grand $L^n$ space introduced in [9]. In particular, we obtain our results with respect to two quantities introduced in [8].