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Volume 19, Issue 2

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

Jacques Giacomoni
• Corresponding author
• CNRS, LMAP (UMR 5142) Bat. IPRA, Université de Pau et des Pays de l’Adour, Avenue de l’Université, 64013 Pau cedex, France
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/ Tuhina Mukherjee
Published Online: 2018-03-21 | DOI: https://doi.org/10.1515/ans-2018-0011

Abstract

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem:

where ${\left(-\mathrm{\Delta }\right)}^{s}$ denotes the fractional Laplace operator for $s\in \left(0,1\right)$, $n>2s$, $q\in \left(0,1\right)$, $\lambda >0$ and Ω is a smooth bounded domain in ${ℝ}^{n}$. Here $f:\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a continuous nondecreasing map satisfying

$\underset{u\to \mathrm{\infty }}{lim}\frac{f\left(u\right)}{{u}^{q+1}}=0.$

We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

MSC 2010: 35R11; 35R09; 35A15

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Accepted: 2018-03-13

Published Online: 2018-03-21

Published in Print: 2019-05-01

The authors were partially funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS 3494 under the project “Singular phenomena in reaction diffusion equations and in conservation laws”.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 2, Pages 333–352, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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