Comparison principle for Singular Fractional $ g- $Laplacian Problems

In this paper, we establish a novel comparison principle of independent interest and prove the uniqueness of weak solutions within the local Orlicz--Sobolev space framework, for the following class of fractional elliptic problems: \begin{equation*} (-\Delta)^{s}_{g} u = f(x) u^{-\alpha} + k(x) u^{\beta}, \quad u > 0 \quad \text{in } \Omega; \quad u = 0 \quad \text{in } \mathbb{R}^{N} \setminus \Omega, \end{equation*} where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, $\alpha > 0$, and $\beta > 0$ satisfies a suitable upper bound. Here, $(-\Delta)^{s}_{g}$ denotes the fractional $g$-Laplacian, with $g$ being the derivative of a Young function $G$. The function $f$ is assumed to be nontrivial, while $k$ is a positive function, and both $f$ and $k$ are assumed to lie in suitable Orlicz spaces. Our analysis relies on a refined variational approach that incorporates a $G$-fractional version of the D\'iaz--Saa inequality together with a $G$-fractional analogue of Picone's identity. These tools, which are of independent interest, also play a key role in the study of simplicity of eigenvalues, Sturmian-type comparison results, Hardy-type inequalities, and related topics.