Sobolev versus H\"older minimizers for the degenerate fractional $p$-Laplacian

Antonio Iannizzotto; Marco Squassina; Sunra Mosconi

arxiv: 1907.08814 · v1 · pith:2CGDKZKKnew · submitted 2019-07-20 · 🧮 math.AP

Sobolev versus H\"older minimizers for the degenerate fractional p-Laplacian

Antonio Iannizzotto , Sunra Mosconi , Marco Squassina This is my paper

Pith reviewed 2026-05-24 18:48 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional p-LaplacianSobolev minimizersHölder minimizersdegenerate caseDirichlet conditionsenergy functionallocal minimizers

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The pith

Local minimizers of the energy for the degenerate fractional p-Laplacian coincide in the fractional Sobolev space and the weighted Hölder space.

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

The paper proves that local minimizers of the energy functional for the fractional p-Laplacian with s between 0 and 1 and p at least 2 are the same whether taken in the fractional Sobolev space W^{s,p}_0(Ω) or in the weighted Hölder space C^0_s(¯Ω). This holds for smooth domains under Dirichlet boundary conditions. A sympathetic reader would care because the two spaces are natural but distinct settings for seeking solutions to this nonlocal degenerate equation, so their minimizers coinciding connects the two approaches directly.

Core claim

We prove that local minimizers of the associated energy functional in the fractional Sobolev space W^{s,p}_0(Ω) and in the weighted Hölder space C^0_s(¯Ω), respectively, do coincide.

What carries the argument

The proven coincidence of local minimizers between W^{s,p}_0(Ω) and C^0_s(¯Ω) for the energy functional of the degenerate fractional p-Laplacian.

If this is right

Any local minimizer found in the Sobolev space is automatically a local minimizer in the Hölder space and vice versa.
Existence or non-existence results for minimizers can be established in either space and carry over to the other.
The Dirichlet problem for the equation can be solved equivalently by minimization in either function space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result may simplify proofs of existence of solutions by allowing the choice of whichever space makes the compactness or lower semicontinuity arguments easier.
It raises the question of whether similar coincidence holds for other nonlocal operators or for global rather than local minimizers.

Load-bearing premise

The domain Ω must be smooth and the parameters must satisfy s in (0,1) with p at least 2.

What would settle it

The existence of a function that is a local minimizer of the energy in one of the two spaces but not in the other would disprove the claim.

read the original abstract

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $\Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(\Omega)$ and in the weighted H\"older space $C^0_s(\overline\Omega)$, respectively, do coincide.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves local minimizers of the energy coincide in W^{s,p}_0 and C^0_s for the degenerate fractional p-Laplacian when p >= 2.

read the letter

The key result is that local minimizers of the energy for the degenerate fractional p-Laplacian coincide when taken in the fractional Sobolev space W^{s,p}_0(Ω) versus the weighted Holder space C^0_s(Ω-bar). This holds under the usual assumptions: smooth bounded domain, s in (0,1), p >= 2, and Dirichlet conditions. The abstract states a direct proof of this equivalence, extending earlier work on non-degenerate or local cases to the degenerate regime. That extension is the main new piece. The paper does a clean job of framing the claim and the parameter range where the operator and spaces are well-defined. It gives researchers working on nonlocal variational problems a choice of function space without changing the set of local minimizers, which can simplify some arguments. The assumptions look standard and do not seem to hide the main difficulty. The degeneracy for p >= 2 is handled in the proof according to the claim, though the abstract gives no steps so the technical details on how the degeneracy is controlled remain to be checked in the full text. No obvious circularity or unjustified transfer between topologies appears in the stated result. This is a technical note aimed at specialists in fractional Sobolev spaces and nonlocal operators. Readers already working on p-Laplacian type equations or regularity for nonlocal equations will find the equivalence useful for choosing the right space in proofs. It is not a broad advance but a solid, focused clarification. The work shows clear engagement with the literature on these spaces and operators. I would send it to peer review; the claim is precise enough and the setting standard enough that referees can assess the details without major setup issues.

Referee Report

0 major / 2 minor

Summary. The paper considers the energy functional associated to the degenerate fractional p-Laplacian (−Δ)_p^s (s∈(0,1), p≥2) with Dirichlet conditions in a smooth bounded domain Ω. It proves that the sets of local minimizers of this functional coincide when the minimization is performed in the fractional Sobolev space W^{s,p}_0(Ω) and in the weighted Hölder space C^0_s(¯Ω), respectively.

Significance. The result establishes an equivalence between local minimizers taken with respect to two distinct topologies on overlapping function spaces. This equivalence is useful for transferring existence, uniqueness, or regularity statements between the Sobolev and Hölder settings for nonlocal degenerate equations. The direct proof (no reduction to auxiliary fitted quantities) and the treatment of the degenerate regime p≥2 are strengths of the work.

minor comments (2)

[§2] §2: the precise definition of the weighted Hölder norm on C^0_s(¯Ω) and its relation to the fractional Sobolev seminorm should be stated explicitly rather than only referenced.
The statement of the main theorem (presumably Theorem 1.1) would benefit from an explicit sentence clarifying that “local minimizer” is understood with respect to the topology of each space.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The report contains no specific major comments, so we have no points requiring a point-by-point reply at this stage.

Circularity Check

0 steps flagged

No significant circularity; direct proof of coincidence of minimizers

full rationale

The paper claims a direct mathematical proof that local minimizers of the energy functional coincide when taken in W^{s,p}_0(Ω) versus C^0_s(Ω̄) under the standard assumptions that make the fractional p-Laplacian and Dirichlet spaces well-defined. No equations, definitions, or steps in the provided abstract or claim reduce by construction to fitted inputs, self-referential definitions, or load-bearing self-citations. The result is presented as a theorem derived from the functional and space properties without renaming known results or smuggling ansatzes. This is a standard non-circular existence/uniqueness-style argument in analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard properties of fractional Sobolev and Hölder spaces and the definition of the fractional p-Laplacian; no free parameters, invented entities, or ad-hoc axioms are apparent from the abstract.

axioms (1)

standard math The fractional p-Laplacian (−Δ)^s_p is well-defined and the associated energy functional admits local minimizers in the stated spaces for s ∈ (0,1) and p ≥ 2
This is the background setup invoked in the abstract for the equation and spaces.

pith-pipeline@v0.9.0 · 5618 in / 1181 out tokens · 27627 ms · 2026-05-24T18:48:04.534881+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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L. Brasco, E. P arini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016) 323–355

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H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris S´ er. I 317 (1993) 465–472

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W. Chen, S. Mosconi, M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018) 3065–3114

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Del Pezzo, A

L.M. Del Pezzo, A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractio nal p- Laplacian, J. Diﬀerential Equations 263 (2017) 765–778

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D ¨uzg¨un, A

F.G. D ¨uzg¨un, A. Iannizzotto, Three nontrivial solutions for nonlinear fractional Lapla cian equations, Adv. Nonlinear Analysis 7 (2018) 211–226

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A. Iannizzotto, S. Mosconi, M. Squassina , Global H¨ older regularity for the fractionalp-Laplacian, Rev. Mat. Iberoam. 32 (2016) 1353–1392

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Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina , Fine boundary regularity for the fractional p-Laplacian, (preprint)

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A. Iannizzotto, N.S. P apageorgiou, Existence of three nontrivial solutions for nonlinear Neum ann hemivariational inequalities, Nonlinear Anal. 70 (2009) 3285–3297

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Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv

S. Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Studies 18 (2018) 691–704

work page 2018
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J. Liu, S. Liu, The existence of multiple solutions to quasilinear ellipti c equations, Bull. London Math. Soc. 37 (2005) 592–600. 14 A. IANNIZZOTTO, S. MOSCONI, M. SQUASSINA

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Mosconi, M

S. Mosconi, M. Squassina , Recent progresses in the theory of nonlinear nonlocal prob lems, Bruno Pini Mathematical Analysis Sem. 7 (2016) 147–164

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Ros-Oton, J

X. Ros-Oton, J. Serra , The Dirichlet problem for the fractional Laplacian: regul arity up to the boundary, J. Math. Pures Appl. 101 (2014) 275–302

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N. S. Ustinov , Multiplicity of positive solutions to the boundary-value problems for fractional Laplacians, J. Math. Sci. 236 (2019), 236: 446

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W ang, On a superlinear elliptic equation, Ann

Z.Q. W ang, On a superlinear elliptic equation, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire8 (1991) 43–57. (A. Iannizzotto) Department of Mathematics and Computer Science Universit`a degli Studi di Cagliari Viale L. Merello 92, 09123 Cagliari, Italy E-mail address : antonio.iannizzotto@unica.it (S. Mosconi) Dipartimento di Matematica e Informatica Unive...

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[1] [1]

Ambrosetti, H

A. Ambrosetti, H. Br ´ezis, G. Cerami, Combined eﬀects of concave-convex nonlinearities in some e lliptic problems, J. Funct. Anal. 122 (1994) 519–543

work page 1994

[2] [2]

Barrios, E

B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire32 (2015) 875–900

work page 2015

[3] [3]

Brasco, E

L. Brasco, E. P arini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016) 323–355

work page 2016

[4] [4]

Brezis, L

H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris S´ er. I 317 (1993) 465–472

work page 1993

[5] [5]

Brock, L

F. Brock, L. Iturriaga, P. Ubilla, A multiplicity result for the p-Laplacian involving a parameter, Ann. Henri Poincar´ e9 (2008) 1371–1386

work page 2008

[6] [6]

W. Chen, S. Mosconi, M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018) 3065–3114

work page 2018

[7] [7]

Del Pezzo, A

L.M. Del Pezzo, A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractio nal p- Laplacian, J. Diﬀerential Equations 263 (2017) 765–778

work page 2017

[8] [8]

D ¨uzg¨un, A

F.G. D ¨uzg¨un, A. Iannizzotto, Three nontrivial solutions for nonlinear fractional Lapla cian equations, Adv. Nonlinear Analysis 7 (2018) 211–226

work page 2018

[9] [9]

F an, On the sub-supersolution method for p(x)-Laplacian equations, J

X. F an, On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007) 665–682

work page 2007

[10] [10]

de Figueiredo, J.P

D.G. de Figueiredo, J.P. Gossez, P. Ubilla, Local ’superlinearity’ and ’sublinearity’ for the p-Laplacian, J. Funct. Anal. 257 (2009) 721–752

work page 2009

[11] [11]

Frassu, Nonlinear Dirichlet problem for the nonlocal anisotropic o perator LK , Comm

S. Frassu, Nonlinear Dirichlet problem for the nonlocal anisotropic o perator LK , Comm. Pure Appl. Anal. 18 (2019) 1847–1867

work page 2019

[12] [12]

Y. Fu, P. Pucci , Multiplicity existence for sublinear fractional Laplaci an problems, Appl. Anal. 96 (2017) 1497–1508

work page 2017

[13] [13]

Garc` ıa Azorero, I

J.P. Garc` ıa Azorero, I. Peral Alonso, J.J. Manfredi, Sobolev versus H¨ older local minimizers and global multiplicity for some quasilinear elliptic equatio ns, Commun. Contemp. Math. 2 (2000) 385–404

work page 2000

[14] [14]

Gasi ´nski, N.S

L. Gasi ´nski, N.S. P apageorgiou, Multiple solutions for nonlinear coercive problems with a n onhomoge- neous diﬀerential operator and a nonsmooth potential, Set-Valued Var. Anal. 20 (2012) 417–443

work page 2012

[15] [15]

Iannizzotto, S

A. Iannizzotto, S. Liu, K. Perera, M. Squassina , Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016) 101–125

work page 2016

[16] [16]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina , H s versus C 0-weighted minimizers, Nonlinear Diﬀer. Equ. Appl. 22 (2015) 477–497

work page 2015

[17] [17]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina , Global H¨ older regularity for the fractionalp-Laplacian, Rev. Mat. Iberoam. 32 (2016) 1353–1392

work page 2016

[18] [18]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina , A note on global regularity for the weak solutions of fractional p-Laplacian equations, Rend. Lincei Mat. Appl. 27 (2016) 15–24

work page 2016

[19] [19]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina , Fine boundary regularity for the fractional p-Laplacian, (preprint)

work page

[20] [20]

Iannizzotto, N.S

A. Iannizzotto, N.S. P apageorgiou, Existence of three nontrivial solutions for nonlinear Neum ann hemivariational inequalities, Nonlinear Anal. 70 (2009) 3285–3297

work page 2009

[21] [21]

Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv

S. Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Studies 18 (2018) 691–704

work page 2018

[22] [22]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic e quations, Nonlinear Anal. 12 (1988) 1203–1219

work page 1988

[23] [23]

J. Liu, S. Liu, The existence of multiple solutions to quasilinear ellipti c equations, Bull. London Math. Soc. 37 (2005) 592–600. 14 A. IANNIZZOTTO, S. MOSCONI, M. SQUASSINA

work page 2005

[24] [24]

Mosconi, M

S. Mosconi, M. Squassina , Recent progresses in the theory of nonlinear nonlocal prob lems, Bruno Pini Mathematical Analysis Sem. 7 (2016) 147–164

work page 2016

[25] [25]

Ros-Oton, J

X. Ros-Oton, J. Serra , The Dirichlet problem for the fractional Laplacian: regul arity up to the boundary, J. Math. Pures Appl. 101 (2014) 275–302

work page 2014

[26] [26]

N. S. Ustinov , Multiplicity of positive solutions to the boundary-value problems for fractional Laplacians, J. Math. Sci. 236 (2019), 236: 446

work page 2019

[27] [27]

W ang, On a superlinear elliptic equation, Ann

Z.Q. W ang, On a superlinear elliptic equation, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire8 (1991) 43–57. (A. Iannizzotto) Department of Mathematics and Computer Science Universit`a degli Studi di Cagliari Viale L. Merello 92, 09123 Cagliari, Italy E-mail address : antonio.iannizzotto@unica.it (S. Mosconi) Dipartimento di Matematica e Informatica Unive...

work page 1991