Weighted anisotropic Sobolev inequality with extremal and associated singular problems

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-07-01 DOI:10.57262/die036-0102-59

K. Bal, Prashanta Garain

引用次数: 5

Abstract

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $\Omega\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_{\Omega}|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_{\Omega}\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(\Omega,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(\Omega,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $\Omega$. We discuss the case $0

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具有极值和相关奇异问题的加权各向异性Sobolev不等式

对于$\mathbb｛R｝^N$中给定的Finsler-Minkowski范数$\mathcal｛F｝$和有界光滑域$\Omega\subet\mathbb{R｝^N$\big（N\geq2\big）$，我们建立了以下加权各向异性Sobolev不等式$S\left（\int_｛\Omega｝|u|^qf\，dx\right）^\frac｛1｝｛p｝，\fquad\fall\，u\在W_0^{1，p}（\Omega，W）\leqno｛\mathcal｛（p）｝$$中，其中$W_0^｛1，p｝（\Omeca，W）$是一类$p$可容许权$W$下的加权Sobolev空间，其中$f$是$\Omega$中的一些非负可积函数。我们讨论了$0

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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