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Turkish Journal of Mathematics

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

First Page

1092

Last Page

1114

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