  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}.
