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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 1
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Research Article

On some singular problems involving the fractional p(x,.) -Laplace operator

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Pages 275-287 | Received 10 Mar 2021, Accepted 26 Jun 2021, Published online: 05 Jul 2021
 

Abstract

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular problem involving the fractional p(x,.)-Laplace operator {(Δ)p(x,.)su+|u|q(x)2u=g(x)uϵ1γ(x)λf(x,u)in Ω,u=0on Ω,where Ω is a smooth bounded domain in RN (N3), 0<s,ϵ<1, λ is a positive parameter and γ:Ω¯(0,ϵ) is a continuous function, p:Ω¯×Ω¯(1,) is a bounded, continuous and symmetric function, q:Ω¯(1,) is a continuous function, gLps(x)ϵps(x)+γ(x)2ϵ(Ω) and g(x)>0 with ps(x)=Np(x,x)Nsp(x,x). Here, the nonlinearity f is in C1(Ω¯×R) and assumed to satisfy suitable assumptions. Using variational methods combined with monotonicity arguments, we obtain the existence of solutions to the problem in a fractional Sobolev space with variable exponent. To our best knowledge, this paper is the first attempt in the study of singular problems involving fractional p(x,.)-Laplace operators.

2010 Mathematics Subject Classifications:

Disclosure statement

No potential conflict of interest was reported by the author(s).

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