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 \( Ω\subset \mathbb{R}^{N} \) is a smooth bounded domain, \( α> 0 \), and \( β> 0 \) satisfies a suitable upper bound. Here, \( (-Δ)^{s}_{g} \) 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 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.