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{equation*} \left\{ \begin{array}{ll} (-\Delta)_{\Phi_{x,y}}^s u = f(x,u),& \mbox{in }\Omega, u=0,& \mbox{on }\mathbb{R}^N\setminus \Omega, \end{array} \right. \end{equation*} where $N\geq 2$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $f:\Omega \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carath\'{e}odory 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.