Extremal functions for the sharp Moser--Trudinger type inequalities in whole space $\mathbb R^N$

TL;DR

This paper investigates sharp Moser-Trudinger inequalities in rica space  N  2, explicitly computes key limits, and proves the existence of maximizers using concentration-compactness, offering an alternative proof to prior results.

Contribution

It explicitly computes normalized vanishing and concentrating limits and proves the existence of maximizers for Moser-Trudinger inequalities in rica space  N  2, extending previous work.

Findings

Explicit formulas for normalized limits of the functional.

Existence of maximizers for the inequalities.

Alternative proof of known existence results.

Abstract

This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized concentrating limit} of the Moser-Trudinger type functional associated with our inequalities over all the \emph{normalized vanishing sequences} and the \emph{normalized concentrating sequences}, respectively. Exploiting these limits together with the concentration-compactness principle of Lions type, we give a proof of the exitence of maximizers for these Moser-Trudinger type inequalities. Our approach gives an alternative proof of the existence of maximizers for the Moser-Trudinger inequality and singular Moser-Trudinger inequality in whole space $\mathbb R^N$ due to Li and Ruf \cite{LiRuf2008} and Li and Yang \cite{LiYang}.