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={}
}