Weighted anisotropic Sobolev inequality with extremal and associated singular problems

IF 1.8 4区数学 Q1 MATHEMATICS

Differential and Integral Equations 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

Differential and Integral Equations MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.40

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Differential and Integral Equations will publish carefully selected research papers on mathematical aspects of differential and integral equations and on applications of the mathematical theory to issues arising in the sciences and in engineering. Papers submitted to this journal should be correct, new, and of interest to a substantial number of mathematicians working in these areas.

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