On the Topology of Fractional Orlicz–Sobolev Spaces and Its Link to Nonlocal g−Laplacian Problems

Sobolev spaces and their implications for nonlinear nonlocal Dirichlet problems governed by
the fractional g−Laplacian. The analysis begins with a detailed investigation of the underlying
structure of the functions N− and Orlicz functions, which constitute the functional setting for
these spaces. Fundamental features such as completeness, separability, reflexivity, and their limiting
behavior as s ↑ 1 are rigorously addressed. Within this setting, a fractional counterpart of the
compact Rellich-Kondrachov embedding theorem is established. As a principal application, the
existence and uniqueness of weak solutions to a non-linear Dirichlet problem are obtained through
a variational approach, relying on monotonicity methods and the Minty–Browder framework. The
results highlight the role of fractional Orlicz–Sobolev spaces in extending the functional analytic
tools required for the treatment of non-local differential models.