Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-w9xp6 Total loading time: 0.89 Render date: 2022-11-30T08:24:41.284Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

Published online by Cambridge University Press:  24 July 2020

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 (marius.ghergu@ucd.ie)
Paschalis Karageorgis
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland (pete@maths.tcd.ie; singhgu@tcd.ie)
Gurpreet Singh
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland (pete@maths.tcd.ie; singhgu@tcd.ie)

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bidaut-Véron, M. F. and Pohozaev, S.. Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84 (2001), 149.CrossRefGoogle Scholar
Caristi, G., D'Ambrosio, L. and Mitidieri, E.. Liouville theorems for some nonlinear inequalities. Proc. Steklov Inst. Math. 260 (2008), 90111.CrossRefGoogle Scholar
Chen, H. and Zhou, F.. Classification of isolated singularities of positive solutions for Choquard equations. J. Differ. Equ. 261 (2016), 66686698.CrossRefGoogle Scholar
Chen, H. and Zhou, F.. Isolated singularities of positive solutions for Choquard equations in sublinear case. Commun. Contemp. Math. 20 (2018), art no 1750040.CrossRefGoogle Scholar
D'Ambrosio, L. and Mitidieri, E.. A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv. Math. 224 (2010), 9670–1020.CrossRefGoogle Scholar
Filippucci, R. and Ghergu, M.. Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms. Nonlinear Anal. 197 (2020), art no 111857.CrossRefGoogle Scholar
Ghergu, M. and Singh, G.. On a class of mixed Choquard–Schrödinger–Poisson systems. Discrete Contin. Dyn. Syst. B 12 (2019), 297309.CrossRefGoogle Scholar
Ghergu, M. and Taliaferro, S.. Pointwise bounds and blow-up for Choquard–Pekar inequalities at an isolated singularity. J. Differ. Equ. 261 (2016), 189217.CrossRefGoogle Scholar
Ghergu, M., Karageorgis, P. and Singh, G.. Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms. J. Differ. Equ. 268 (2020), 60336066.CrossRefGoogle Scholar
Lieb, E. and Loss, M.. Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn (Rhode-Island USA: Amer., Math. Soc., 2001).Google Scholar
Liskevich, V., Lyakhova, S. and Moroz, V.. Positive solutions to nonlinear $p$-Laplace equations with Hardy potential in exterior domains. J. Differ. Equ. 232 (2007), 212252.CrossRefGoogle Scholar
Mitidieri, E. and Pohozaev, S. I.. A priori estimates and blow up of solutions to nonlinear partial differential equations. Proc. Steklov Inst. Math. 234 (2001), 1367.Google Scholar
Moroz, V. and Van Schaftingen, J.. Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains. J. Differ. Equ. 254 (2013), 30893145.CrossRefGoogle Scholar
Moroz, V. and Van Schaftingen, J.. A guide to the Choquard equation. J. Fixed: Point Theory Appl. 19 (2017), 773813.Google Scholar
Singh, G.. Nonlocal perturbations of the fractional Choquard equation. Adv. Nonlinear Anal. 8 (2019), 694706.CrossRefGoogle Scholar
Wang, Y. and Hajaiej, H.. Boundary singularity of Choquard equation in the half space. J. Math. Pures Appl. 126 (2019), 232248.CrossRefGoogle Scholar
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *