An improved Trudinger--Moser inequality and its extremal functions involving $L^p$-norm in $\mathbb{R}^2$

Authors

XIAOMENG LI

DOI

10.3906/mat-1907-24

Abstract

Let $W^{1,2}(\mathbb{R}^2)$ be the standard Sobolev space. Denote for any real number $p>2$ \begin{align*}\lambda_{p}=\inf\limits_{u\in W^{1,2}(\mathbb{R}^2),u\not\equiv0}\frac{\int_{\mathbb{R}^{2}}( \nabla u ^2+ u ^2)dx}{(\int_{\mathbb{R}^{2}} u ^pdx)^{2/p}}. \end{align*} Define a norm in $W^{1,2}(\mathbb{R}^2)$ by \begin{align*}\ u\ _{\alpha,p}=\left(\int_{\mathbb{R}^{2}}( \nabla u ^2+ u ^2)dx-\alpha(\int_{\mathbb{R}^{2}} u ^pdx)^{2/p}\right)^{1/2}\end{align*} where $0\leq\alpha2$ and $0\leq\alpha

Keywords

Trudinger--Moser inequality, extremal function, blow-up analysis

First Page

1092

Last Page

1114

Recommended Citation

LI, XIAOMENG (2020) "An improved Trudinger--Moser inequality and its extremal functions involving $L^p$-norm in $\mathbb{R}^2$," Turkish Journal of Mathematics: Vol. 44: No. 4, Article 2. https://doi.org/10.3906/mat-1907-24
Available at: https://journals.tubitak.gov.tr/math/vol44/iss4/2