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*} (-Δ)^{s}_{g} u = f(x) u^{-α} + k(x) u^β, \quad u > 0 \quad $\text{in }$ Ω; \quad u = 0 \quad $\text{in }$ $\mathbb{R}$^{N} \setminus Ω, \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $\alpha >0$, and $\beta >0$ satisfies a suitable upper bound. Here, $(-\mathrm{\Delta }{)}_g^s$ 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 Díaz–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.