Abdelhamid Gouasmia (2022) Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity. Math Meth Appl Sci. , 45(9), 5283-5303
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Abstract
In the present paper, we investigate the following singular quasilinear elliptic system:
\\\\begin{equation*}\\\\tag{S}
\\\\left\\\\lbrace
\\\\begin{aligned}
\\\\left( -\\\\Delta \\\\right)_{p_{1}} ^{s_{1}}u&=\\\\dfrac{1}{u^{\\\\alpha _{1}}\\\\,v^{\\\\beta _{1}}}, \\\\, &u > 0 \\\\quad \\\\text{in}\\\\, \\\\Omega ;\\\\quad &u = 0, \\\\quad \\\\text{in }\\\\,\\\\mathbb{R}^{N}\\\\setminus \\\\Omega,\\\\vspace{0.2cm}\\\\\\\\
\\\\left( -\\\\Delta \\\\right)_{p_{2}} ^{s_{2}}v&=\\\\dfrac{1}{v^{\\\\alpha _{2}}\\\\,u^{\\\\beta _{2}}}, \\\\, &v > 0 \\\\quad \\\\text{in}\\\\, \\\\Omega ;\\\\quad &v = 0, \\\\quad \\\\text{in }\\\\,\\\\mathbb{R}^{N}\\\\setminus \\\\Omega,
\\\\end{aligned}\\\\right.
\\\\end{equation*}
where $ \\\\Omega\\\\subset \\\\mathbb{R}^{N} $ is an open bounded domain with smooth boundary, $s_{1}, s_{2} \\\\in (0,1) ,$ $ p_{1}, p_{2} \\\\in(1, +\\\\infty) $ and $\\\\alpha _{1},\\\\,\\\\alpha _{2},\\\\,\\\\beta _{1},\\\\,\\\\beta _{2}$ are positive constants. We first discuss the non-existence of positive classical solutions to system $\\\\text{(S)}.$
%Next, we prove existence, uniqueness and H\\\\\\\"{o}lder regularity of positive weak solutions.
Next, constructing suitable ordered pairs of sub- and supersolutions, we apply Schauder’s Fixed Point Theorem in the associated conical shell and get the existence of a positive weak solutions pair to $\\\\text{(S)}$, turn to be H\\\\\\\"older continuous. Finally, we apply a well-known Krasnoselski\\\\v{i}\\\'s argument to establish the uniqueness of such positive pair of solutions.
\\\\begin{equation*}\\\\tag{S}
\\\\left\\\\lbrace
\\\\begin{aligned}
\\\\left( -\\\\Delta \\\\right)_{p_{1}} ^{s_{1}}u&=\\\\dfrac{1}{u^{\\\\alpha _{1}}\\\\,v^{\\\\beta _{1}}}, \\\\, &u > 0 \\\\quad \\\\text{in}\\\\, \\\\Omega ;\\\\quad &u = 0, \\\\quad \\\\text{in }\\\\,\\\\mathbb{R}^{N}\\\\setminus \\\\Omega,\\\\vspace{0.2cm}\\\\\\\\
\\\\left( -\\\\Delta \\\\right)_{p_{2}} ^{s_{2}}v&=\\\\dfrac{1}{v^{\\\\alpha _{2}}\\\\,u^{\\\\beta _{2}}}, \\\\, &v > 0 \\\\quad \\\\text{in}\\\\, \\\\Omega ;\\\\quad &v = 0, \\\\quad \\\\text{in }\\\\,\\\\mathbb{R}^{N}\\\\setminus \\\\Omega,
\\\\end{aligned}\\\\right.
\\\\end{equation*}
where $ \\\\Omega\\\\subset \\\\mathbb{R}^{N} $ is an open bounded domain with smooth boundary, $s_{1}, s_{2} \\\\in (0,1) ,$ $ p_{1}, p_{2} \\\\in(1, +\\\\infty) $ and $\\\\alpha _{1},\\\\,\\\\alpha _{2},\\\\,\\\\beta _{1},\\\\,\\\\beta _{2}$ are positive constants. We first discuss the non-existence of positive classical solutions to system $\\\\text{(S)}.$
%Next, we prove existence, uniqueness and H\\\\\\\"{o}lder regularity of positive weak solutions.
Next, constructing suitable ordered pairs of sub- and supersolutions, we apply Schauder’s Fixed Point Theorem in the associated conical shell and get the existence of a positive weak solutions pair to $\\\\text{(S)}$, turn to be H\\\\\\\"older continuous. Finally, we apply a well-known Krasnoselski\\\\v{i}\\\'s argument to establish the uniqueness of such positive pair of solutions.
Information
| Item Type | Journal |
|---|---|
| Divisions | |
| ePrint ID | 5247 |
| Date Deposited | 2024-12-17 |
| Further Information | Google Scholar |
| URI | https://univ-soukahras.dz/en/publication/article/5247 |
BibTex
@article{uniusa5247,
title={Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity},
author={Abdelhamid Gouasmia},
journal={Math Meth Appl Sci.}
year={2022},
volume={45},
number={9},
pages={5283-5303},
publisher={}
}
title={Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity},
author={Abdelhamid Gouasmia},
journal={Math Meth Appl Sci.}
year={2022},
volume={45},
number={9},
pages={5283-5303},
publisher={}
}