On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent

Anouar Bahrouni; VicenŢiu D. RĂdulescu

doi:10.3934/dcdss.2018021

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2018, Volume 11, Issue 3: 379-389. Doi: 10.3934/dcdss.2018021

This issue Previous Article Entire solutions of nonlocal elasticity models for composite materials Next Article Global compactness results for nonlocal problems

On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent

Anouar Bahrouni^1, and
VicenŢiu D. RĂdulescu^2,3, ,

1.
Mathematics Department, University of Monastir, Faculty of Sciences, 5019 Monastir, Tunisia
2.
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13,200585 Craiova, Romania

* Corresponding author: Vicent¸iu D. Rădulescu
* Corresponding author: Vicent¸iu D. Rădulescu

Received: April 30, 2017

Revised: July 31, 2017

Published: June 2018

The second author is supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0130.

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Abstract

The content of this paper is at the interplay between function spaces $L^{p(x)}$ and $W^{k, p(x)}$ with variable exponents and fractional Sobolev spaces $W^{s, p}$. We are concerned with some qualitative properties of the fractional Sobolev space $W^{s, q(x), p(x, y)}$, where $q$ and $p$ are variable exponents and $s∈ (0, 1)$. We also study a related nonlocal operator, which is a fractional version of the nonhomogeneous $p(x)$-Laplace operator. The abstract results established in this paper are applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J91, 35S30, 46E35, 58E30.

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