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arxiv: 2605.09170 · v2 · submitted 2026-05-09 · 🧮 math.AP

On singular problems in nonreflexive fractional Orlicz-Sobolev spaces

Marcos L.M. Carvalho , Luana C. M. Lima , Carlos A.P. Santos , Maxwell L. Silva This is my paper

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Existence and uniqueness of positive solutions for singular fractional Orlicz-Sobolev problems with convergence to the local case as s approaches 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work studies equations with a singular right-hand side that blows up when the solution is zero. It works in fractional Orlicz-Sobolev spaces that lack reflexivity, so usual compactness arguments do not apply directly. The authors minimize an energy functional and construct special test functions to show that the minimizer satisfies the equation in a weak sense. They also prove that these fractional solutions approach the solution of the corresponding non-fractional problem when the fractional parameter s tends to 1. The approach avoids relying on reflexivity by using a direct minimization strategy tailored to the singular term.

Core claim

Existence and uniqueness of positive solution u_s for the singular quasilinear problem (−Δ_Φ)^s u = u^{-γ} in the nonreflexive fractional Orlicz-Sobolev W^s_0 L^Φ(Ω) for 0<s<1. Furthermore, u_s converges in L^Φ(Ω) to the unique positive solution u in W^1_0 L^Φ(Ω) of −Δ_Ψ u = u^{-γ} as s ↑ 1.

Load-bearing premise

The energy functionals are not well-defined on the whole space due to lack of reflexivity and the singular term; the new test-function construction is assumed to overcome this and prove that positive minimizers are weak solutions.

read the original abstract

In this work, we deal with existence and uniqueness of positive solution $u_s$ for the singular quasilinear problem $(-\Delta_{\Phi})^su=u^{-\gamma}$ in the nonreflexive fractional Orlicz-Sobolev $ W^{s}_0L^{\Phi}(\Omega)$ for $0<s<1$. Furthermore, we show that $u_s$ converges in $L^{\Phi}(\Omega)$ to the unique positive solution $u\in W^{1}_0L^{\Phi}(\Omega)$ of the problem $-\Delta_{\Psi}u=u^{-\gamma}$ as $s \uparrow 1$, where $\Psi$ is an appropriate $N$-function equivalent to the $N$-function $\Phi$. The main difficulties to obtain existence of weak solutions for both singular quasilinear problems are that their associate energy functionals may not be well-defined on their whole natural workspaces due to the lack of the reflexivity and the presence of the singular term. To overcome these difficulties, we will use the minimization method and present a new approach to building appropriate test functions to prove that the problems have positive minimizers that we showed to be weak solutions of them, respectively.

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Referee Report

2 major / 2 minor

Summary. The manuscript establishes existence and uniqueness of positive weak solutions u_s to the singular problem (−Δ_Φ)^s u = u^{-γ} in the nonreflexive space W^s_0 L^Φ(Ω) for 0 < s < 1. It proceeds by minimization of an energy functional, combined with a new construction of test functions that is claimed to show the positive minimizers are weak solutions despite the functional not being well-defined on the whole space (due to nonreflexivity and the singular term). The paper further proves that u_s converges in L^Φ(Ω) to the unique positive solution u of the local problem −Δ_Ψ u = u^{-γ} in W^1_0 L^Φ(Ω) as s ↑ 1, where Ψ is an N-function equivalent to Φ.

Significance. If the test-function construction is fully rigorous, the work would be a useful contribution to singular quasilinear problems in nonreflexive fractional Orlicz-Sobolev spaces, where the direct method fails for lack of weak compactness. The convergence result linking the fractional and local cases would also be of interest for understanding the limit behavior of these operators.

major comments (2)
  1. [Proof of existence for the fractional problem] The central existence argument rests on the new test-function construction (detailed after the minimization step in the fractional case). It is not shown explicitly how these test functions remain admissible in the nonreflexive setting and simultaneously justify both the existence of a minimizer and passage to the weak form of the equation; without weak lower-semicontinuity or compactness, an additional verification that the construction circumvents these issues is required.
  2. [Section on the limit as s ↑ 1] In the convergence proof as s ↑ 1, the modular convergence in L^Φ(Ω) is used to pass to the limit in the singular term, but the estimates controlling the difference between the fractional and local energies (via the equivalence of Φ and Ψ) are not quantified sufficiently to guarantee that the limit u satisfies the local equation without additional regularity assumptions on the minimizers.
minor comments (2)
  1. [Abstract] In the abstract, 'associate energy functionals' should read 'associated energy functionals'.
  2. [Introduction] The notation for the N-functions Φ and Ψ and their relation to the modulars could be introduced with a short preliminary subsection to improve readability for readers outside the immediate Orlicz-space community.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of N-functions Φ and Ψ being equivalent and appropriate, plus the assumption that the new test functions suffice to verify the weak formulation despite missing reflexivity.

axioms (1)
  • standard math N-functions Φ and Ψ satisfy standard growth and equivalence conditions for Orlicz-Sobolev spaces
    Invoked to define the spaces W^s_0 L^Φ(Ω) and the local limit space.

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