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Extremal functions for the sharp Moser--Trudinger type inequalities in whole space $\mathbb R^N$. (arXiv:1705.05864v1 [math.FA])
来源于:arXiv
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}. 查看全文>>