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On Fractional Musielak–Sobolev Spaces and Applications to Nonlocal Problems

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Abstract

In this work, we establish some abstract results on the perspective of the fractional Musielak–Sobolev spaces, such as: uniform convexity, Radon–Riesz property with respect to the modular function, \((S_{+})\)-property, Brezis–Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems

$$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta )_{\varPhi _{x,y}}^s u = f(x,u),&{} \text{ in } \varOmega ,\\ u=0,&{} \text{ on } {\mathbb {R}}^N\setminus \varOmega , \end{array} \right. \end{aligned}$$

where \(N\ge 2\), \(\varOmega \subset {\mathbb {R}}^N\) is a bounded domain with Lipschitz boundary \(\partial \varOmega \) and \(f:\varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function not necessarily satisfying the Ambrosetti–Rabinowitz condition. Such class of problems enables the presence of many particular operators, for instance, the fractional operator with variable exponent, double-phase and double-phase with variable exponent operators, anisotropic fractional p-Laplacian, among others.

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Acknowledgements

The first and third authors were partially supported by CNPq with Grants 304699/2021-7 and 316643/2021-1, respectively. The second author was partially supported by CAPES.

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Authors and Affiliations

  1. Universidade Federal de Pernambuco, Recife, Brazil

    J. C. de Albuquerque & L. R. S. de Assis

  2. Universidade Federal de Goiás, Goiânia, Brazil

    M. L. M. Carvalho

  3. Universidad de Buenos Aires, Buenos Aires, Argentina

    A. Salort

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  1. J. C. de Albuquerque
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  2. L. R. S. de Assis
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  3. M. L. M. Carvalho
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  4. A. Salort
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de Albuquerque, J.C., de Assis, L.R.S., Carvalho, M.L.M. et al. On Fractional Musielak–Sobolev Spaces and Applications to Nonlocal Problems. J Geom Anal 33, 130 (2023). https://doi.org/10.1007/s12220-023-01211-2

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