Sharp Subcritical and Critical Trudinger-Moser Inequalities on \(\mathbb {R}^{2}\) and their Extremal Functions

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• 作者：Nguyen Lam
• 关键词：Trudinger ; Moser inequality ; Unbounded domains ; Critical growth ; Extremal function ; Sharp constants ; Concentration compactness.
• 刊名：Potential Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：46
• 期：1
• 页码：75-103
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Potential Theory; Probability Theory and Stochastic Processes; Geometry; Functional Analysis;
• 出版者：Springer Netherlands
• ISSN：1572-929X
• 卷排序：46

文摘

In this paper, we study on \(\mathbb {R}^{2}\) some new types of the sharp subcritical and critical Trudinger-Moser inequality that have close connections to the study of the optimizers for the classical Trudinger-Moser inequalities. For instance, one of our results can be read as follows: Let 0 ≤ β < 2, p ≥ 0, α ≥ 0. Then$$\sup_{\left\Vert \nabla u\right\Vert_{2}^{2}+\left\Vert u\right\Vert_{2} ^{2}\leq1}\left\Vert u\right\Vert_{2}^{p}{\int}_{\mathbb{R}^{2}}\exp\left( \alpha\left( 1-\frac{\beta}{2}\right) \left\vert u\right\vert^{2}\right) \left\vert u\right\vert^{2}\frac{dx}{\left\vert x\right\vert^{\beta}}<\infty $$if and only if α < 4π or α = 4π, p ≥ 2. The attainability and inattainability of these sharp inequalties will be also investigated using a new approach, namely the relations between the supremums of the sharp subcritical and critical ones. This new method will enable us to compute explicitly the supremums of the subcritical Trudinger-Moser inequalities in some special cases. Also, a version of Concentration-compactness principle in the spirit of Lions ( Lions, I. Rev. Mat. Iberoam. 1(1) 145–01 1985) will also be studied.KeywordsTrudinger-Moser inequalityUnbounded domainsCritical growthExtremal functionSharp constantsConcentration compactness.Research of this work was partly supported by an AMS-Simons Travel Grant.