On Fractional Musielak–Sobolev Spaces and Applications to Nonlocal Problems

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.