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Advanced Nonlinear Studies

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Volume 19, Issue 2

Issues

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 / Konijeti Sreenadh
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:

{(-Δ)su=λf(u)uq,u>0 in Ω,u=0in nΩ,

where (-Δ)s denotes the fractional Laplace operator for s(0,1), n>2s, q(0,1), λ>0 and Ω is a smooth bounded domain in n. Here f:[0,)[0,) is a continuous nondecreasing map satisfying

limuf(u)uq+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.

Keywords: Fractional Laplacian; Singular Nonlinearity; Infinite Semipositone Problem; Sub-Supersolutions; Three Positive Solutions

MSC 2010: 35R11; 35R09; 35A15

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About the article


Received: 2018-01-19

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, DOI: https://doi.org/10.1515/ans-2018-0011.

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