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Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  24 July 2020

Marius Ghergu ,
Paschalis Karageorgis  and
Gurpreet Singh
Marius Ghergu
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland Institute of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei St., Bucharest, 010702, Romania ([email protected])
Paschalis Karageorgis
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland ([email protected]; [email protected])
Gurpreet Singh
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland ([email protected]; [email protected])
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Abstract

We study the quasilinear elliptic inequality

\[ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, \]
where $p>0$, $q, \mu \in \mathbb {R}$, $m>1$ and $I_\alpha$ is the Riesz potential of order $\alpha \in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.

Keywords

Quasilinear elliptic inequalitiesm-Laplace operatorHardy term

MSC classification

Primary: 35J62: Quasilinear elliptic equations
Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 1075 - 1093
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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