On a class of fractional systems with nonstandard growth conditions

Boumazourh, Athmane; Azroul, Elhoussine

doi:10.1007/s11868-019-00310-5

Cited by 11 publications

(3 citation statements)

References 15 publications

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“…The fractional Sobolev spaces with variable exponents were first introduced recently by Kaufmann, Rossi, and Vidal [43], and have been studied in different contexts. See [4, 6–15, 19, 25, 39, 40, 50, 55] and references therein. Note that the Triebel–Lizorkin spaces with variable smoothness and integrability have been introduced in [30], which are isomorphic to

W^{k, p(\cdot )}(\mathbb {R}^n)

k \in \mathbb {N} \cup \lbrace 0 \rbrace

, respectively, the variable exponent Bessel potential space

\mathcal {L}^{\alpha , p(\cdot )}(\mathbb {R}^n)

for

\alpha &gt; 0

under suitable assumptions on p .…”

Section: Preliminariesmentioning

confidence: 99%

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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W^{k, p(\cdot )}(\mathbb {R}^n)

k \in \mathbb {N} \cup \lbrace 0 \rbrace

, respectively, the variable exponent Bessel potential space

\mathcal {L}^{\alpha , p(\cdot )}(\mathbb {R}^n)

for

\alpha &gt; 0

under suitable assumptions on p .…”

Section: Preliminariesmentioning

confidence: 99%

Local regularity for nonlocal equations with variable exponents

Chaker

Kim

2023

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…The fractional Sobolev spaces with variable exponents were first introduced recently by Kaufmann, Rossi, and Vidal [43], and have been studied in different contexts. See [25,11,12,4,7,10,39,8,13,15,19,50,6,9,14,40,55] and references therein. Note that the Triebel-Lizorkin spaces with variable smoothness and integrability have been introduced in [31], which are isomorphic to…”

Section: Preliminariesmentioning

confidence: 99%

Local regularity for nonlocal equations with variable exponents

Chaker¹,

Kim²

2021

Preprint

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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-Höldertype 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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“…These results correspond to the "sublinear" and the "superlinear" cases. Our paper is motivated by [4,8,13,28] and organized as follows: in Section 2 we recall some notations and properties of fractional Lebesgue and Sobolev spaces with variable exponents. In order to prove the main theorems in Section 3 some useful lemmas are given.…”

Section: Introductionmentioning

confidence: 99%

Existence of Solutions for a Class of Fractional Kirchhoff-type Systems in $\mathbb{R}^N$ with Non-standard Growth

2021

Self Cite

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This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in R N involving the fractional pseudodifferential operators defined as the generalizations of the p(x)-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory.The obtained results of this note significantly contribute to the study of Kirchhofftype systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

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On a class of fractional systems with nonstandard growth conditions

Cited by 11 publications

References 15 publications

Local regularity for nonlocal equations with variable exponents

Local regularity for nonlocal equations with variable exponents

Local regularity for nonlocal equations with variable exponents

Existence of Solutions for a Class of Fractional Kirchhoff-type Systems in $\mathbb{R}^N$ with Non-standard Growth

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