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 {(-Δ)Φx,ysu=f(x,u),inΩ,u=0,onRNΩ,where N≥ 2 , Ω⊂ RN is a bounded domain with Lipschitz boundary ∂Ω and f: Ω× R→ 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.