On fractional critical problems with multi-polar Hardy potentials

Debangana Mukherjee; Edoardo Mainini; Roberto Ognibene

arxiv: 2604.27492 · v1 · submitted 2026-04-30 · 🧮 math.AP

On fractional critical problems with multi-polar Hardy potentials

Edoardo Mainini , Debangana Mukherjee , Roberto Ognibene This is my paper

Pith reviewed 2026-05-07 09:31 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Laplacianmulti-polar Hardy potentialSobolev critical exponentconcentration-compactness principleexistence of minimizersasymptotic estimatesextended formulation

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The pith

The existence of minimizers for fractional critical equations with multi-polar Hardy potentials is determined by the masses at the poles and the distances between them.

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

This paper studies positive solutions to fractional equations that feature both a multi-polar Hardy potential and a Sobolev critical nonlinearity. The nonlocal character of the fractional Laplacian and the lack of explicit ground states for the single-pole case create significant technical challenges. The authors address these by reformulating the problem in an extended local setting and deriving precise asymptotic estimates for single-pole solutions, which they then use in a concentration-compactness argument to establish conditions for the existence of minimizers. A reader would care because these criteria clarify when such doubly critical problems admit solutions, with applications to models involving nonlocal diffusion and singular potentials.

Core claim

Through an extended formulation and sharp asymptotic estimates for the single-pole problem, the authors apply a concentration-compactness principle to show that minimizers exist when the magnitudes of the masses and the mutual distances between the poles satisfy appropriate relations.

What carries the argument

The concentration-compactness argument relying on sharp asymptotic estimates transferred from the single-pole extended formulation.

Load-bearing premise

The sharp asymptotic estimates for single-pole solutions are accurate enough to control the concentration-compactness analysis in the multi-pole case.

What would settle it

Constructing a specific multi-pole configuration where a minimizer exists despite violating the mass-distance criterion, or vice versa.

read the original abstract

We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of explicit ground states for the single-pole equation stand as major difficulties. We overcome these obstacles by passing to an extended formulation of the problem and by establishing sharp asymptotic estimates for the solutions in the case of a single pole. Then, through a concentration-compactness argument, we show that the existence of minimizers is dictated by the magnitude of the masses and the mutual distances between the corresponding poles.

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 extends single-pole fractional critical Hardy results to the multi-polar case by deriving sharp asymptotics in the extension formulation and running concentration-compactness to tie existence to masses and pole distances.

read the letter

The main point is that the authors handle the double criticality of multi-polar Hardy potentials plus Sobolev critical nonlinearity for the fractional Laplacian. They move to the extended formulation to get around the lack of explicit single-pole ground states, establish sharp decay estimates there, and then use concentration-compactness to show that minimizing sequences are compact precisely when the masses and mutual distances satisfy a certain threshold condition. This produces an existence criterion that depends on those quantities rather than just the total mass. The approach is a straightforward but non-trivial adaptation of single-pole techniques, and the abstract indicates they carry the estimates through to control the cross terms from multiple poles. That is the concrete advance. The work is technically honest in its framing of the obstacles and the tools used to address them. The potential soft spot is exactly the one raised in the stress-test note: the single-pole asymptotics must have remainders small enough that the interaction energies stay below the critical level when poles interact. If the error is only O(1/|x|^α) with α too small relative to s, the comparison argument could fail for certain distance regimes. The paper claims to overcome this, so the proofs need to be checked for the precise orders and constants. No circularity appears in the strategy, and the methods are drawn from the existing literature without obvious gaps in the outline. This is for people already working on nonlocal variational problems with singular potentials and critical exponents. A reader who knows the single-pole extension method and concentration-compactness arguments will follow the extension without trouble and can assess the new estimates directly. It is worth sending to peer review so that experts can verify the asymptotic control and the resulting existence statement.

Referee Report

2 major / 2 minor

Summary. The manuscript addresses the existence of positive solutions to a fractional critical elliptic equation featuring a multi-polar Hardy potential and a Sobolev-critical nonlinearity. The authors pass to an extended local formulation, derive sharp asymptotic expansions for the single-pole problem, and apply a concentration-compactness argument to obtain an existence criterion that depends on the masses at the poles and the distances between them.

Significance. If the asymptotic estimates and the ensuing compactness analysis hold, the result would provide a precise, distance-dependent existence threshold for a nonlocal problem with two sources of criticality, a setting where explicit solutions are unavailable and standard compactness fails. The combination of extension technique with controlled remainder terms in the single-pole expansion is a technically substantive contribution that could serve as a template for related multi-singularity problems.

major comments (2)

[§3] §3 (single-pole asymptotics): the claimed expansion u(x) ∼ c/|x−x0|^{N−2s} + lower-order terms must be accompanied by an explicit remainder estimate whose decay is strictly faster than the interaction terms that appear when two or more poles are present; otherwise the comparison of the mountain-pass value against the single-pole threshold (used in §4) cannot be justified uniformly in the distances.
[§4.2] §4.2 (concentration-compactness): the proof that a minimizing sequence remains compact when the mutual distances exceed a critical threshold relies on the single-pole energy being strictly less than the multi-pole energy; this comparison must be verified with the precise error terms from §3, not merely asserted via the abstract form of the expansion.

minor comments (2)

[Introduction] The notation for the extended variable and the Caffarelli–Silvestre extension operator should be recalled briefly in the introduction for readers who may not have the reference at hand.
[§4] A short table summarizing the dependence of the existence threshold on the number of poles and their configuration would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The observations regarding the need for explicit remainder estimates and their use in the energy comparison have led us to strengthen the presentation in §§3 and 4. We address each major comment below.

read point-by-point responses

Referee: §3 (single-pole asymptotics): the claimed expansion u(x) ∼ c/|x−x0|^{N−2s} + lower-order terms must be accompanied by an explicit remainder estimate whose decay is strictly faster than the interaction terms that appear when two or more poles are present; otherwise the comparison of the mountain-pass value against the single-pole threshold (used in §4) cannot be justified uniformly in the distances.

Authors: We agree that an explicit rate on the remainder is required to control the comparison uniformly in the inter-pole distances. In the revised manuscript we have strengthened Theorem 3.1 by deriving the expansion in the extended formulation with a quantified remainder: the error term satisfies |R(x)| ≤ C |x − x_0|^{-(N−2s + δ)} for an explicit δ > 0 depending only on N and s. This decay is strictly faster than the leading interaction terms of order d^{-(N−2s)}, where d is the distance between poles. The proof of the mountain-pass level comparison in §4 now invokes this rate directly, ensuring the argument holds uniformly for all distances above the critical threshold. A new remark following Theorem 3.1 explains the comparison with the interaction scale. revision: yes
Referee: §4.2 (concentration-compactness): the proof that a minimizing sequence remains compact when the mutual distances exceed a critical threshold relies on the single-pole energy being strictly less than the multi-pole energy; this comparison must be verified with the precise error terms from §3, not merely asserted via the abstract form of the expansion.

Authors: We concur that the energy comparison must be carried out with the concrete error terms rather than the abstract expansion. In the revised §4.2 we have inserted a new auxiliary lemma (Lemma 4.3) that substitutes the explicit remainder from the updated Theorem 3.1 into the difference between the multi-pole energy and the sum of single-pole energies. The resulting estimate shows that, whenever the distances exceed the critical value determined by the pole masses, the multi-pole functional value lies strictly above the single-pole threshold. This quantitative gap rules out dichotomy and yields compactness of minimizing sequences. The original concentration-compactness outline is retained, but the verification step is now fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the derivation chain

full rationale

The paper derives sharp asymptotic estimates for the single-pole problem in the extended formulation as an independent step, then transfers those estimates into a concentration-compactness argument for the multi-pole case to obtain a distance- and mass-dependent existence criterion. This does not reduce any prediction to a fitted input by construction, nor does it rely on self-definitional loops, load-bearing self-citations, or smuggled ansatzes. The single-pole analysis precedes and supports the multi-pole analysis without feedback, and the abstract and description indicate reliance on external analytic tools rather than internal redefinition. The central claim therefore retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the work implicitly rests on standard background results in fractional Sobolev spaces, Hardy inequalities, and concentration-compactness principles whose details are not visible here.

pith-pipeline@v0.9.0 · 5387 in / 1134 out tokens · 70563 ms · 2026-05-07T09:31:09.930522+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Brasco and G

L. Brasco and G. Franzina , Convexity properties of Dirichlet integrals and Picone -type inequalities, Kodai Math. J. 37 no. 3 (2014), 769--799 (English). doi:10.2996/kmj/1414674621

work page doi:10.2996/kmj/1414674621 2014
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Cabr´ e and Y

X. Cabr\' e and Y. Sire , Nonlinear equations for fractional L aplacians, I : R egularity, maximum principles, and H amiltonian estimates, Ann. Inst. H. Poincar\' e Anal. Non Lin\' e aire 31 no. 1 (2014), 23--53. 3165278 . doi:10.1016/j.anihpc.2013.02.001

work page doi:10.1016/j.anihpc.2013.02.001 2014
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An extension problem related to the fractional

L. Caffarelli and L. Silvestre , An extension problem related to the fractional L aplacian, Comm. Partial Differential Equations 32 no. 7-9 (2007), 1245--1260. 2354493 . doi:10.1080/03605300600987306

work page doi:10.1080/03605300600987306 2007
[4]

Chan , M

H. Chan , M. D. M. Gonz \'a lez , Y. Huang , E. Mainini , and B. Volzone , Uniqueness of entire ground states for the fractional plasma problem, Calc. Var. Partial Differ. Equ. 59 no. 6 (2020), 41 (English), Id/No 195. doi:10.1007/s00526-020-01845-y

work page doi:10.1007/s00526-020-01845-y 2020
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Cinti , R

E. Cinti , R. Ognibene , and B. Ruffini , A quantitative stability inequality for fractional capacities, Math. Eng. (Springfield) 4 no. 5 (2022), 28 (English), Id/No 44. doi:10.3934/mine.2022044

work page doi:10.3934/mine.2022044 2022
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Cotsiolis and N

A. Cotsiolis and N. K. Tavoularis , Best constants for S obolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 no. 1 (2004), 225--236. 2064421 . doi:10.1016/j.jmaa.2004.03.034

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Dipierro , M

S. Dipierro , M. Medina , and E. Valdinoci , Fractional elliptic problems with critical growth in the whole of R ^n , Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 15, Edizioni della Normale, Pisa, 2017. 3617721 . doi:10.1007/978-88-7642-601-8

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Dipierro , L

S. Dipierro , L. Montoro , I. Peral , and B. Sciunzi , Qualitative properties of positive solutions to nonlocal critical problems involving the H ardy- L eray potential, Calc. Var. Partial Differential Equations 55 no. 4 (2016), Art. 99, 29. 3528440 . doi:10.1007/s00526-016-1032-5

work page doi:10.1007/s00526-016-1032-5 2016
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Journal of Mathematical Analysis and Applications 47(2), 324–353 (1974)

I. Ekeland , On the variational principle, J. Math. Anal. Appl. 47 (1974), 324--353 (English). doi:10.1016/0022-247X(74)90025-0

work page doi:10.1016/0022-247x(74)90025-0 1974
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M. M. Fall , Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential, J. Funct. Anal. 264 no. 3 (2013), 661--690 (English). doi:10.1016/j.jfa.2012.09.021

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M. M. Fall and T. Weth , Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 no. 8 (2012), 2205--2227 (English). doi:10.1016/j.jfa.2012.06.018

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Felli , On the existence of ground state solutions to nonlinear Schr \"o dinger equations with multisingular inverse-square anisotropic potentials, J

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work page doi:10.1007/s11854-009-0023-2 2009
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Felli , E

V. Felli , E. M. Marchini , and S. Terracini , On S chr\" o dinger operators with multipolar inverse-square potentials, J. Funct. Anal. 250 no. 2 (2007), 265--316. 2352482 . doi:10.1016/j.jfa.2006.10.019

work page doi:10.1016/j.jfa.2006.10.019 2007
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Felli , E

V. Felli , E. M. Marchini , and S. Terracini , On Schr \"o dinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. J. 58 no. 2 (2009), 617--676 (English). doi:10.1512/iumj.2009.58.3471

work page doi:10.1512/iumj.2009.58.3471 2009
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Felli , D

V. Felli , D. Mukherjee , and R. Ognibene , On fractional multi-singular S chr\" o dinger operators: positivity and localization of binding, J. Funct. Anal. 278 no. 4 (2020), 108389, 47. 4044748 . doi:10.1016/j.jfa.2019.108389

work page doi:10.1016/j.jfa.2019.108389 2020
[18]

Felli and S

V. Felli and S. Terracini , Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 no. 1-3 (2006), 469--495. 2209764 . doi:10.1080/03605300500394439

work page doi:10.1080/03605300500394439 2006
[19]

Felli and S

V. Felli and S. Terracini , Nonlinear Schr \"o dinger equations with symmetric multi-polar potentials, Calc. Var. Partial Differ. Equ. 27 no. 1 (2006), 25--58 (English). doi:10.1007/s00526-006-0020-6

work page doi:10.1007/s00526-006-0020-6 2006
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R. L. Frank , E. H. Lieb , and R. Seiringer , Hardy- L ieb- T hirring inequalities for fractional S chr\" o dinger operators, J. Amer. Math. Soc. 21 no. 4 (2008), 925--950. 2425175 . doi:10.1090/S0894-0347-07-00582-6

work page doi:10.1090/s0894-0347-07-00582-6 2008
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R. L. Frank and R. Seiringer , Non-linear ground state representations and sharp H ardy inequalities, J. Funct. Anal. 255 no. 12 (2008), 3407--3430. 2469027 . doi:10.1016/j.jfa.2008.05.015

work page doi:10.1016/j.jfa.2008.05.015 2008
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I. W. Herbst , Spectral theory of the operator (p^ 2 +m^ 2 )^ 1/2 -Ze^ 2 /r , Comm. Math. Phys. 53 no. 3 (1977), 285--294. 0436854 . Available at http://projecteuclid.org/euclid.cmp/1103900706

work page arXiv 1977
[23]

E. H. Lieb , The stability of matter: From atoms to stars, Bull. Am. Math. Soc., New Ser. 22 no. 1 (1990), 1--49 (English). doi:10.1090/S0273-0979-1990-15831-8

work page doi:10.1090/s0273-0979-1990-15831-8 1990
[24]

E. H. Lieb and M. Loss , Stability of a model of relativistic quantum electrodynamics, Commun. Math. Phys. 228 no. 3 (2002), 561--588 (English). doi:10.1007/s002200200665

work page doi:10.1007/s002200200665 2002
[25]

Mainini , R

E. Mainini , R. Ognibene , and B. Volzone , Local multiplicity for fractional linear equations with Hardy potentials, Calc. Var. Partial Differ. Equ. 64 no. 2 (2025), 20 (English), Id/No 51. doi:10.1007/s00526-024-02914-2

work page doi:10.1007/s00526-024-02914-2 2025
[26]

S. A. Molchanov and E. Ostrovskij , Symmetric stable processes as traces of degenerate diffusion processes, Theory Probab. Appl. 14 (1969), 128--131 (English). doi:10.1137/1114012

work page doi:10.1137/1114012 1969
[27]

Terracini , On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv

S. Terracini , On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ. 1 no. 2 (1996), 241--264 (English)

work page 1996

[1] [1]

Brasco and G

L. Brasco and G. Franzina , Convexity properties of Dirichlet integrals and Picone -type inequalities, Kodai Math. J. 37 no. 3 (2014), 769--799 (English). doi:10.2996/kmj/1414674621

work page doi:10.2996/kmj/1414674621 2014

[2] [2]

Cabr´ e and Y

X. Cabr\' e and Y. Sire , Nonlinear equations for fractional L aplacians, I : R egularity, maximum principles, and H amiltonian estimates, Ann. Inst. H. Poincar\' e Anal. Non Lin\' e aire 31 no. 1 (2014), 23--53. 3165278 . doi:10.1016/j.anihpc.2013.02.001

work page doi:10.1016/j.anihpc.2013.02.001 2014

[3] [3]

An extension problem related to the fractional

L. Caffarelli and L. Silvestre , An extension problem related to the fractional L aplacian, Comm. Partial Differential Equations 32 no. 7-9 (2007), 1245--1260. 2354493 . doi:10.1080/03605300600987306

work page doi:10.1080/03605300600987306 2007

[4] [4]

Chan , M

H. Chan , M. D. M. Gonz \'a lez , Y. Huang , E. Mainini , and B. Volzone , Uniqueness of entire ground states for the fractional plasma problem, Calc. Var. Partial Differ. Equ. 59 no. 6 (2020), 41 (English), Id/No 195. doi:10.1007/s00526-020-01845-y

work page doi:10.1007/s00526-020-01845-y 2020

[5] [5]

Cinti , R

E. Cinti , R. Ognibene , and B. Ruffini , A quantitative stability inequality for fractional capacities, Math. Eng. (Springfield) 4 no. 5 (2022), 28 (English), Id/No 44. doi:10.3934/mine.2022044

work page doi:10.3934/mine.2022044 2022

[6] [6]

Cotsiolis and N

A. Cotsiolis and N. K. Tavoularis , Best constants for S obolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 no. 1 (2004), 225--236. 2064421 . doi:10.1016/j.jmaa.2004.03.034

work page doi:10.1016/j.jmaa.2004.03.034 2004

[7] [7]

Dipierro , M

S. Dipierro , M. Medina , and E. Valdinoci , Fractional elliptic problems with critical growth in the whole of R ^n , Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 15, Edizioni della Normale, Pisa, 2017. 3617721 . doi:10.1007/978-88-7642-601-8

work page doi:10.1007/978-88-7642-601-8 2017

[8] [8]

Dipierro , L

S. Dipierro , L. Montoro , I. Peral , and B. Sciunzi , Qualitative properties of positive solutions to nonlocal critical problems involving the H ardy- L eray potential, Calc. Var. Partial Differential Equations 55 no. 4 (2016), Art. 99, 29. 3528440 . doi:10.1007/s00526-016-1032-5

work page doi:10.1007/s00526-016-1032-5 2016

[9] [9]

Journal of Mathematical Analysis and Applications 47(2), 324–353 (1974)

I. Ekeland , On the variational principle, J. Math. Anal. Appl. 47 (1974), 324--353 (English). doi:10.1016/0022-247X(74)90025-0

work page doi:10.1016/0022-247x(74)90025-0 1974

[10] [10]

M. M. Fall , Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential, J. Funct. Anal. 264 no. 3 (2013), 661--690 (English). doi:10.1016/j.jfa.2012.09.021

work page doi:10.1016/j.jfa.2012.09.021 2013

[11] [11]

M. M. Fall , Semilinear elliptic equations for the fractional Laplacian with Hardy potential, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 193 (2020), 29 (English), Id/No 111311. doi:10.1016/j.na.2018.07.008

work page doi:10.1016/j.na.2018.07.008 2020

[12] [12]

M. M. Fall and V. Felli , Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 no. 2 (2014), 354--397. 3169789 . doi:10.1080/03605302.2013.825918

work page doi:10.1080/03605302.2013.825918 2014

[13] [13]

M. M. Fall and T. Weth , Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 no. 8 (2012), 2205--2227 (English). doi:10.1016/j.jfa.2012.06.018

work page doi:10.1016/j.jfa.2012.06.018 2012

[14] [14]

Felli , On the existence of ground state solutions to nonlinear Schr \"o dinger equations with multisingular inverse-square anisotropic potentials, J

V. Felli , On the existence of ground state solutions to nonlinear Schr \"o dinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math. 108 (2009), 189--217 (English). doi:10.1007/s11854-009-0023-2

work page doi:10.1007/s11854-009-0023-2 2009

[15] [15]

Felli , E

V. Felli , E. M. Marchini , and S. Terracini , On S chr\" o dinger operators with multipolar inverse-square potentials, J. Funct. Anal. 250 no. 2 (2007), 265--316. 2352482 . doi:10.1016/j.jfa.2006.10.019

work page doi:10.1016/j.jfa.2006.10.019 2007

[16] [16]

Felli , E

V. Felli , E. M. Marchini , and S. Terracini , On Schr \"o dinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. J. 58 no. 2 (2009), 617--676 (English). doi:10.1512/iumj.2009.58.3471

work page doi:10.1512/iumj.2009.58.3471 2009

[17] [17]

Felli , D

V. Felli , D. Mukherjee , and R. Ognibene , On fractional multi-singular S chr\" o dinger operators: positivity and localization of binding, J. Funct. Anal. 278 no. 4 (2020), 108389, 47. 4044748 . doi:10.1016/j.jfa.2019.108389

work page doi:10.1016/j.jfa.2019.108389 2020

[18] [18]

Felli and S

V. Felli and S. Terracini , Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations 31 no. 1-3 (2006), 469--495. 2209764 . doi:10.1080/03605300500394439

work page doi:10.1080/03605300500394439 2006

[19] [19]

Felli and S

V. Felli and S. Terracini , Nonlinear Schr \"o dinger equations with symmetric multi-polar potentials, Calc. Var. Partial Differ. Equ. 27 no. 1 (2006), 25--58 (English). doi:10.1007/s00526-006-0020-6

work page doi:10.1007/s00526-006-0020-6 2006

[20] [20]

R. L. Frank , E. H. Lieb , and R. Seiringer , Hardy- L ieb- T hirring inequalities for fractional S chr\" o dinger operators, J. Amer. Math. Soc. 21 no. 4 (2008), 925--950. 2425175 . doi:10.1090/S0894-0347-07-00582-6

work page doi:10.1090/s0894-0347-07-00582-6 2008

[21] [21]

R. L. Frank and R. Seiringer , Non-linear ground state representations and sharp H ardy inequalities, J. Funct. Anal. 255 no. 12 (2008), 3407--3430. 2469027 . doi:10.1016/j.jfa.2008.05.015

work page doi:10.1016/j.jfa.2008.05.015 2008

[22] [22]

I. W. Herbst , Spectral theory of the operator (p^ 2 +m^ 2 )^ 1/2 -Ze^ 2 /r , Comm. Math. Phys. 53 no. 3 (1977), 285--294. 0436854 . Available at http://projecteuclid.org/euclid.cmp/1103900706

work page arXiv 1977

[23] [23]

E. H. Lieb , The stability of matter: From atoms to stars, Bull. Am. Math. Soc., New Ser. 22 no. 1 (1990), 1--49 (English). doi:10.1090/S0273-0979-1990-15831-8

work page doi:10.1090/s0273-0979-1990-15831-8 1990

[24] [24]

E. H. Lieb and M. Loss , Stability of a model of relativistic quantum electrodynamics, Commun. Math. Phys. 228 no. 3 (2002), 561--588 (English). doi:10.1007/s002200200665

work page doi:10.1007/s002200200665 2002

[25] [25]

Mainini , R

E. Mainini , R. Ognibene , and B. Volzone , Local multiplicity for fractional linear equations with Hardy potentials, Calc. Var. Partial Differ. Equ. 64 no. 2 (2025), 20 (English), Id/No 51. doi:10.1007/s00526-024-02914-2

work page doi:10.1007/s00526-024-02914-2 2025

[26] [26]

S. A. Molchanov and E. Ostrovskij , Symmetric stable processes as traces of degenerate diffusion processes, Theory Probab. Appl. 14 (1969), 128--131 (English). doi:10.1137/1114012

work page doi:10.1137/1114012 1969

[27] [27]

Terracini , On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv

S. Terracini , On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ. 1 no. 2 (1996), 241--264 (English)

work page 1996