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arXiv · 2509.01783

Embedding Results and Fractional $p(x,.)$-Laplacian Problem in Unbounded Domains

Abdelkrim Barbara·Ahmed Bousmaha·Mohammed Shimi
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Abstract

In this paper, we prove a new continuous embedding theorem for fractional Sobolev spaces with variable exponents into variable exponent Lebesgue spaces on unbounded domains. As an application, we study a class of nonlocal elliptic problems driven by the fractional $p(x, \cdot)$-Laplacian operator. Using variational methods combined with the established embedding result, we prove the existence of nontrivial weak solutions under suitable growth and regularity conditions on the nonlinearity.\\ A significant analytical challenge addressed in this work arises from both the nonlocal behavior of the fractional $p(x, \cdot)$-Laplacian and the lack of compactness induced by the unboundedness of the domain.

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Abdelkrim Barbara, Ahmed Bousmaha, Mohammed Shimi. 2025-09-01. Embedding Results and Fractional $p(x,.)$-Laplacian Problem in Unbounded Domains. https://arxiv.org/abs/2509.01783

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