Jacques Giacomoni, Abdelhamid Gouasmia and Abdelhafid Mokrane (2022) Discrete Picone inequalities and applications to non local and non homogenenous operators. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. , 116(100),
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Abstract
In this article, we prove new discrete Picone inequalities, associated to non local elliptic operators as the fractional $ p- $Laplace operator, denoted by $\\left( -\\Delta \\right) ^{s}_{p} u$ and defined as:
$$ \\left( -\\Delta \\right) ^{s}_{p} u(x) := 2\\, \\text{\\textbf{P.V.}} \\int_{\\mathbb{R}^{N}}\\dfrac{\\left| u(x) - u(y)\\right| ^{p-2}(u(x) - u(y))}{\\left| x-y\\right|^{N + ps} }\\,dy $$
where $ p>1, $ $ 0 < s < 1 $ and \\textbf{P.V.} denotes the Cauchy principal value. As applications, we provide new results about existence, non-existence and uniqueness of weak positive solutions to problems involving fractional and non homogeneous operators as $ (-\\Delta)_{p}^{s_{1}} + (-\\Delta)_{q}^{s_{2}}$, where $ s_{1}, s_{2} \\in (0,1) $ and $ 1 < q, p<\\infty$. In addition, for this class of operators, we obtain comparison principles, a Sturmian comparison principle and a Hardy-type inequality with weight. Finally, we also establish some qualitative results for nonlinear and non local elliptic systems with sub-homogeneous growth.
$$ \\left( -\\Delta \\right) ^{s}_{p} u(x) := 2\\, \\text{\\textbf{P.V.}} \\int_{\\mathbb{R}^{N}}\\dfrac{\\left| u(x) - u(y)\\right| ^{p-2}(u(x) - u(y))}{\\left| x-y\\right|^{N + ps} }\\,dy $$
where $ p>1, $ $ 0 < s < 1 $ and \\textbf{P.V.} denotes the Cauchy principal value. As applications, we provide new results about existence, non-existence and uniqueness of weak positive solutions to problems involving fractional and non homogeneous operators as $ (-\\Delta)_{p}^{s_{1}} + (-\\Delta)_{q}^{s_{2}}$, where $ s_{1}, s_{2} \\in (0,1) $ and $ 1 < q, p<\\infty$. In addition, for this class of operators, we obtain comparison principles, a Sturmian comparison principle and a Hardy-type inequality with weight. Finally, we also establish some qualitative results for nonlinear and non local elliptic systems with sub-homogeneous growth.
Information
| Item Type | Journal |
|---|---|
| Divisions | |
| ePrint ID | 5246 |
| Date Deposited | 2024-12-17 |
| Further Information | Google Scholar |
| URI | https://univ-soukahras.dz/en/publication/article/5246 |
BibTex
@article{uniusa5246,
title={Discrete Picone inequalities and applications to non local and non homogenenous operators},
author={Jacques Giacomoni, Abdelhamid Gouasmia and Abdelhafid Mokrane},
journal={Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.}
year={2022},
volume={116},
number={100},
pages={},
publisher={}
}
title={Discrete Picone inequalities and applications to non local and non homogenenous operators},
author={Jacques Giacomoni, Abdelhamid Gouasmia and Abdelhafid Mokrane},
journal={Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.}
year={2022},
volume={116},
number={100},
pages={},
publisher={}
}