Strauss and Lions Type Theorems for the Fractional Sobolev Spaces with Variable Exponent and Applications to Nonlocal Kirchhoff–Choquard Problem

Bahrouni, Sabri; Ounaies, Hichem

doi:10.1007/s00009-020-01661-w

Cited by 15 publications

(6 citation statements)

References 33 publications

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“…Moreover, the Musielak-Sobolev space W s,Gx,y (Ω) becomes the fractional Sobolev space with variable exponent W s,p(•,•) (R d ) and the fractional Musielak g x,y -Laplace operator turns into the fractional p(x, y)-Laplacian. Therefore, our results (Theorems 1.1, 1.2, 1.3, 1.4, 1.5, and 1.6) remain valid for fractional Sobolev space with variable exponent which are related to the main results shown in [19,30,32]. It is clear that the generalized N-function G x,y satisfies the assumptions (g 1 ) − (g 5 ) and (B f ).…”

Section: Some Examplessupporting

confidence: 68%

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

Bahrouni¹,

Hlel²,

Ounaies³

2023

Preprint

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This paper deals with new continuous and compact embedding theorems for the fractional Musielak-Sobolev spaces in R d . As an application, using the variational methods, we obtain the existence of nontrivial weak solution for the following Schrödinger equationWe would like to mention that the theory of the fractional Musielak-Sobolev spaces is in a developing state and there are few papers in this topic, see [6,11,12]. Note that, all these latter works dealt with bounded case and there are no results devoted for the fractional Musielak-Sobolev spaces in R d . Since the embedding results are crucial in applying variational methods, this work will provide a bridge between the fractional Mueislak-Sobolev theory and PDE's.

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Section: Some Examplessupporting

confidence: 68%

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

Bahrouni¹,

Hlel²,

Ounaies³

2023

Preprint

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“…Assume (G ′ 1 ) and let f : R → R be a continuous function satisfying (f 1 )-(f 2 ). Let (u n ) ⊂ W s,G (R N ) a sequence satisfying the condition (8). Then…”

Section: Proof Of Theorem 16mentioning

confidence: 99%

“…Strauss [19] was the first who observed that there exists an interplay between the regularity of the function and its radially symmetric property. Later on, Strauss's result was generalized in many directions, see [1,2,8,9,12,14,15,21] for a survey of related results and elementary proofs of some of them within the framework of different spaces.…”

Section: Introductionmentioning

confidence: 99%

Problems involving the fractional $g$-Laplacian with Lack of Compactness

Bahrouni¹,

Ounaies²,

Elfalah³

2022

Preprint

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In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space W s,G R N consisting of radial functions, our target embedding spaces are of Orlicz type. Also, we prove a Lions and Lieb type results for W s,G R N that works together in a particular way to get a sequence whose the weak limit is non trivial. As an application, we study the existence of solutions to Quasilinear elliptic problems in the whole space R N involving the fractional g−Laplacian operator, where the conjugated function G of G doesn't satisfy the ∆ 2 -condition.

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“…Especially, for Choquard-Kirchhoff equations with variable exponent in [18], Bahrouni et al dealt with Strauss and Lions type theorems and studied the existence and multiplicity of weak solutions. Furthermore, for nonlocal Choquard-Kirchhoff problems in [19], Biswas et al obtained the existence of ground state solution, and infinitely many weak solutions, which the conditions for nonlinear functions are weaker than the Ambrosetti-Rabinowitz conditions.…”

Section: Introductionmentioning

confidence: 99%

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

et al. 2021

Mathematics

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In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

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Strauss and Lions Type Theorems for the Fractional Sobolev Spaces with Variable Exponent and Applications to Nonlocal Kirchhoff–Choquard Problem

Cited by 15 publications

References 33 publications

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

On the fractional Musielak-Sobolev spaces in R^d: Embedding results & applications

Problems involving the fractional $g$-Laplacian with Lack of Compactness

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

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