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

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