On Fractional Musielak-Sobolev spaces and applications to nonlocal problems

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{equation*} \left{ \begin{array}{ll} (-Δ){Φ{x,y}}^s u = f(x,u),& \mbox{in }Ω, u=0,& \mbox{on }$\mathbb{R}$^N\setminus Ω, \end{array} \right. \end{equation*} where $N\ge 2$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega }$ and $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ 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.