On a class of fractional <i>p</i>(<i>x</i>) -Kirchhoff type problems

Azroul, Elhoussine; Benkirane, Abdelmoujib; Shimi, Mohammed; Srati, Mohammed

doi:10.1080/00036811.2019.1603372

Cited by 52 publications

(32 citation statements)

References 34 publications

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“…Our main goal in this paper is to extend the fractional Sobolev spaces with variable exponents to cover the nonlocal general case with singular kernel. For this, we begin this work by remembering the definition of fractional Sobolev spaces with variable exponent, see for instance [3,4,11,15].…”

Section: Introductionmentioning

confidence: 99%

General fractional Sobolev space with variable exponent and applications to nonlocal problems

2020

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In this paper, we extend the fractional Sobolev spaces with variable exponents W s,p(x,y) to include the general fractional case W K,p(x,y) , where p is a variable exponent, s ∈ (0, 1) and K is a suitable kernel. We are concerned with some qualitative properties of the space W K,p(x,y) (completeness, reflexivity, separability and density). Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discus the existence of a nontrivial solution for a nonlocal p(x, .)-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type L p(x,.) K . 2010 Mathematics Subject Classification. 46E35, 35R11, 35S05, 35J35.

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Section: Introductionmentioning

confidence: 99%

General fractional Sobolev space with variable exponent and applications to nonlocal problems

2020

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“…More precisely, the problems involving the fractional p-Laplacian operator (−∆) s p have been investigated by many papers, see for example [5,11,14] and the references therein. More recently, there are many researchers extended the constant case to include the fractional variable exponent case [2,3,4,6,9,16] where the authors introduced some definitions and basic properties of new fractional Sobolev spaces with variable exponents and obtained some existence results for related nonlocal fractional problems with variable exponents by means of different variational methods. For our first goal, we aim to keep on the study of such nonlocal problems with variable exponent by considering the fractional p(x, .…”

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“…In order to study problem (P s ), some properties on variable exponent Lebesgue spaces and Sobolev spaces, L q(x) (Ω) and W s,p(x,y) (Q) respectively, are required and listed below. We refer the reader to [12,17] and [2,3,6,16] for exhaustive details on properties of those spaces. Moreover, we give several important properties of p(x, .…”

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On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition

Azroul¹,

Benkirane²,

Shimi³

2021

DCDS-S

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The present paper deals with the existence and multiplicity of solutions for a class of fractional p(x, .)-Laplacian problems with the nonlocal Dirichlet boundary data, where the nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.

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Local regularity for nonlocal equations with variable exponents

Chaker

Kim

2023

Mathematische Nachrichten

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In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler-Lagrange equations. We show that weak solutions are locally bounded when the variable exponent 𝑝 is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on 𝑝. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder-type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on 𝑝 outside the domain.

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On a class of fractional p(x) -Kirchhoff type problems

Cited by 52 publications

References 34 publications

General fractional Sobolev space with variable exponent and applications to nonlocal problems

General fractional Sobolev space with variable exponent and applications to nonlocal problems

On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition

Local regularity for nonlocal equations with variable exponents

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