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

2017/05/16 by Van Hoang Nguyen, Nguyen, Van Hoang
Mathematics · #Spectral Theory in Mathematical Physics #Nonlinear Partial Differential Equations #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.1705.05864

Abstract

This paper is devoted to study the sharp Moser-Trudinger type inequalities in\nwhole space mathbb R^N, N \≥ 2 in more general case. We first compute\nexplicitly the \normalized vanishing limit and the \normalized\nconcentrating limit of the Moser-Trudinger type functional associated with our\ninequalities over all the \normalized vanishing sequences and the\n\normalized concentrating sequences, respectively. Exploiting these\nlimits together with the concentration-compactness principle of Lions type, we\ngive a proof of the exitence of maximizers for these Moser-Trudinger type\ninequalities. Our approach gives an alternative proof of the existence of\nmaximizers for the Moser-Trudinger inequality and singular Moser-Trudinger\ninequality in whole space mathbb R^N due to Li and Ruf citeLiRuf2008 and\nLi and Yang citeLiYang.\n

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