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Mixed Anisotropic and Nonlocal Sobolev Type Inequalities with Extremal

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We establish mixed anisotropic and nonlocal Sobolev type inequalities with an extremal. We show that the extremal function is unique up to a multiplicative constant that is associated with the corresponding mixed anisotropic and nonlocal singular partial differential equation. We prove that such a mixed Sobolev type inequality is necessary and sufficient for the existence of solutions to the associated mixed anisotropic and nonlocal singular partial differential equation. We establish a relation between the purely anisotropic singular equation and mixed anisotropic and nonlocal singular equation.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research Berhampur, Berhampur, Odisha, 760010, India

    Prashanta Garain

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  1. Prashanta Garain
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Correspondence to Prashanta Garain.

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Garain, P. Mixed Anisotropic and Nonlocal Sobolev Type Inequalities with Extremal. J Math Sci 281, 633–645 (2024). https://doi.org/10.1007/s10958-024-07141-w

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  • DOI: https://doi.org/10.1007/s10958-024-07141-w

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