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<p>The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, etc.) are established, including the continuous and compact embedding results. Moreover, some functional properties of anisotropic fractional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\vec {p}(.,.)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>p</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-Laplacian operator are proved. As an application, we use the mountain pass theorem and Ekeland’s variational principle to ensure the existence of a weak solution for a nonlocal anisotropic problem with variable exponent. The results presented in this work offer significant insights into the structure and properties of anisotropic fractional Sobolev spaces, laying a robust foundation for future investigations in related fields.</p>

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Anisotropic fractional Sobolev spaces with variable exponent and application to nonlocal problems

  • 期刊论文

  • 发表日期：

    2025年7月5日

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  • E. Azroul,
  • A. Barbara,
  • N. Kamali,
  • M. Shimi

摘要

The main goal of this paper is to introduce a new fractional anisotropic Sobolev space with variable exponent where the basic qualitative properties (completeness, separability, reflexivity, etc.) are established, including the continuous and compact embedding results. Moreover, some functional properties of anisotropic fractional \(\vec {p}(.,.)\) p → ( . , . ) -Laplacian operator are proved. As an application, we use the mountain pass theorem and Ekeland’s variational principle to ensure the existence of a weak solution for a nonlocal anisotropic problem with variable exponent. The results presented in this work offer significant insights into the structure and properties of anisotropic fractional Sobolev spaces, laying a robust foundation for future investigations in related fields.