Some existence results for a nonlocal non-isotropic problem

IF 1 Q1 MATHEMATICS
R. Bentifour, Sofiane El-Hadi Miri
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引用次数: 4

Abstract

In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt \gamma \lt 1.\)
一类非局部非各向同性问题的存在性结果
本文处理以下问题\[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\],其中\(\Omega\)是\(\mathbb{R}^{N}\)中的有界正则域。在不失一般性的前提下,我们假设\(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\)、\(f\)和\(g\)是属于合适勒贝格空间\(L^{m}(\Omega)\)、\(1\lt q\lt \overline{p}^{\ast}\)、\(a\gt 0\)、\(b\gt 0\)和的非负函数 \(0\lt \gamma \lt 1.\)
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
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