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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.21185 |
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| _version_ | 1866918106711982080 |
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| author | Gouasmia, Abdelhamid Bal, Kaushik |
| author_facet | Gouasmia, Abdelhamid Bal, Kaushik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparison principle for Singular Fractional $ g- $Laplacian Problems Gouasmia, Abdelhamid Bal, Kaushik Analysis of PDEs 35J75, 35R11, 35J62 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. |
| title | Comparison principle for Singular Fractional $ g- $Laplacian Problems |
| topic | Analysis of PDEs 35J75, 35R11, 35J62 |
| url | https://arxiv.org/abs/2507.21185 |