Regularity results for a class of mixed local and nonlocal singular problems involving distance function

Kaushik Bal; Stuti Das

REVIEW 1 major objections 4 minor 2 cited by

Regularity results for a class of mixed local and nonlocal singular problems involving distance function

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves up-to-boundary Hölder and gradient-Hölder regularity for the singular mixed local-nonlocal problem $-\Delta_p u+(-\Delta)_q^s u=f(x)u^{-\delta}$ with $f$ blowing up like $d^{-\beta}$.

desk verdict The global Hölder claim for β+δ>1 is not proved because the local regularity constants blow up near the boundary; the regular-problem results and existence theory are solid enough to merit peer review. read the letter →

arxiv 2411.14217 v2 pith:IWTZHPB7 submitted 2024-11-21 math.AP

Kaushik Bal , Stuti Das This is my paper · ORCID

classification math.AP MSC 35J7535M1035R11

keywords Mixedlocal-nonlocalequationHölderregularitySingularnonlinearityDistancefunctionExistenceanduniquenessFractionalp-LaplacianBoundarybehaviorQuasilinearelliptic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies positive zero-outside solutions of $-\Delta_p u+(-\Delta)_q^s u=f(x)u^{-\delta}$ in a bounded $C^2$ domain, where $11$, with local gradient-Hölder regularity in all these cases. The paper thereby gives the doubly singular problem a coherent boundary-regularity picture despite the non-homogeneous mixed operator.

What carries the argument

The central object is the distance function $d(x)=\mathrm{dist}(x,\partial\Omega)$, together with barrier functions of the form $w(x)=\Gamma(d(x)+\varepsilon^{1/\tau})^\tau$, where $\tau=(p-\beta)/(p-1+\delta)$. Because $\partial\Omega$ is $C^2$, $d$ is smooth in a boundary neighbourhood, and the paper computes the action of the $p$-Laplacian on these powers exactly, while showing through estimates for fractional powers of distance that the fractional $q$-Laplacian of the truncated barrier is controlled. These barriers produce the boundary behavior of the approximating solutions and support the weak comparison principle. In the interior, gradient Hölder regularity is obtained by comparison with solutions $h$ of a frozen homogeneous problem on each ball; the difference $|u-h|$ is controlled by energy estimates and a nonlocal tail functional that records the contribution of $u$ away from the ball. The singular weight parameter $\beta$ enters through a weighted integrability condition, which is why the comparison principle is stated only for $\beta<2-1/p$.

What would settle it

Take $\Omega$ to be the unit ball, $p=q=2$, $s=1/2$, and $f=d^{-\beta}$ with $\beta+\delta>1$; Theorem 2.31 predicts $u\in C^{0,(2-\beta)/(1+\delta)}(\Omega)$. Computing the boundary quotient $\limsup_{x\to\partial\Omega} u(x)/d(x)^{(2-\beta)/(1+\delta)}$ for the constructed solution—positive and finite if the predicted boundary behavior is exact, zero or infinite if it is not—would settle the boundary-regularity claim.

Watch

Extended reading notes

Core claim

The central claim, Theorem 2.31, is that for $\beta\in[0,p)$ and $\delta>0$, the weak solution constructed by approximation—or the unique solution when the comparison principle applies—belongs to $C^{1,\sigma}(\Omega)$ for some $\sigma\in(0,1)$ if $\beta+\delta<1$; to $C^{0,\eta}(\Omega)$ for every $\eta\in(0,1)$ if $\beta+\delta=1$; and, if $\beta+\delta>1$, to $C^{0,(p-\beta)/(p-1+\delta)}(\Omega)$ except in the case $\beta=p-q's(p-1+\delta)$, where it belongs to $C^{0,(p-\beta_1)/(p-1+\delta)}(\Omega)$ for every $\beta_1\in(\beta,p)$, with $q'=q/(q-1)$. In the last three cases the solution is also $C^{1,\gamma}$ in the interior. These results rest on a systematic regularity theory for the nonsingular operator $\mathcal{L}u=-\operatorname{div}A(x,\nabla u)+\text{fractional nonlocal term}$: local Hölder regularity when the right-hand side is in $L^n_{\mathrm{loc}}$, local gradient Hölder regularity when it is in $L^d_{\mathrm{loc}}$ with $d>n$, and a boundary $C^{1,\gamma}$ theorem under $0\le f\le Cd^{-\sigma}$ with $\sigma<1$ together with a one-sided bound $0\le u\le Cd^{\epsilon}$. The singular solution is obtained as the increasing limit of solutions with $u^{-\delta}$ replaced by $(u+\varepsilon)^{-\delta}$, and its boundary behavior is pinned down by distance barriers as $u\asymp d^{(p-\beta)/(p-1+\delta)}$, with different powers when $\beta+\delta\le1$.

Load-bearing premise

The load-bearing premise for the uniqueness and comparison part is that the singular weight is integrable enough at the boundary for a weighted inequality of Hardy type to apply, and this holds only when $\beta<2-1/p$; if that condition fails, the proof still gives existence and regularity but does not select a unique solution.

Editorial extensions

If this is right

When the weight is bounded at the boundary ($\beta=0$) and $\delta<1$, every weak solution is $C^{1,\sigma}$ up to the boundary, not merely interior.
At $\beta+\delta=1$ the solution is Hölder continuous up to the boundary with every exponent below $1$, while its gradient is Hölder continuous in the interior; this almost-Lipschitz boundary regularity is obtained without requiring the solution to be in the energy space.
For $\beta+\delta>1$ the boundary Hölder exponent equals the power appearing in the boundary behavior $u\asymp d^{(p-\beta)/(p-1+\delta)}$, so the Hölder and boundary-behavior statements are mutually consistent.
Existence is sharp in $\beta$: no weak solution exists for $\beta\ge p$, and the Sobolev-regularity theorem says exactly when $u$, or a power $u^\theta$, belongs to $W^{1,p}_0(\Omega)$.
As a direct application, the singular perturbed problem $-\Delta_p u+(-\Delta)_q^s u=\lambda u^{-\delta}+b(x,u)$ with critical growth in $b$ has solutions in $C^{1,\sigma}(\Omega)$ when $\delta<1$ and $\beta=0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exponent $(p-\beta)/(p-1+\delta)$ is likely sharp: the proved lower and upper distance bounds have exactly this power, and boundary Hölder regularity cannot in general improve beyond the power governing the boundary behavior; a radial model would make this testable.
The restriction $\beta<2-1/p$ in the uniqueness statement probably reflects the method rather than the phenomenon, since existence and boundary behavior are established for all $\beta<p$; a comparison argument that avoids the weighted integrability step might extend uniqueness to the full range.
The interior regularity estimates are developed for a broad class of operators and for solutions only locally in $W^{1,p}$, so they could be reused for other singular, critical, or lower-order problems without repeating the distance-barrier construction.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague —

The headline: the paper’s main global-Hölder claim for the singular range β+δ>1 (Theorem 2.31(iii)–(iv)) is not actually proved. The two-case argument in §7.4 breaks down because the local Hölder constant from Theorem 2.19(a) depends on ‖f u^{−δ}‖_{L^n(B_R)}, and for balls at distance R from ∂Ω that norm behaves like R^{1−β−δζ} with ζ=(p−β)/(p−1+δ), which blows up as R→0 whenever β+δ>1. The boundary bound u≤Cd^ζ controls values but not oscillations at scales below R. A boundary Campanato estimate with the singular weight d^{−β−δζ} is missing. So the advertised up-to-boundary C^{0,ζ} regularity is unsubstantiated as written; the case β+δ=1 is borderline and may be okay.

That said, the paper is not a throwaway. The regular-problem results (Theorems 2.17–2.21) genuinely extend the De Filippis–Mingione and Antonini–Cozzi machinery to a nonhomogeneous (p,q) mixed operator with solutions in a weaker local class, and those look solid. The existence theory for the singular problem (Theorem 2.27), the boundary behavior in Definition 2.26, and the optimal Sobolev regularity are coherent and useful, and the comparison principle in Theorem 2.28 is a real achievement in the restricted range β<2−1/p. The authors are honest about the restrictions in the theorems; the problem is the abstract, which promises existence and uniqueness without them.

Other soft spots: the proofs lean heavily on [24] and [1], which makes verification slow, but that is standard practice, not a flaw. The uniqueness limitation means Theorem 2.31’s phrase “the unique solution” only applies in the restricted parameter range; for larger β the constructed solution is not known to be unique.

If I were the editor, I’d send it to peer review rather than desk-reject: the topic is active, the regular results have independent value, and the gap in §7.4 is identifiable and potentially fixable. The referee should demand either a genuine boundary Campanato estimate or a clear explanation of why the local constants don’t blow up. The abstract also needs to match the theorem statements.

For a reading group: maybe — worth discussing the gap, but I wouldn’t present it as a finished global regularity result.

Referee Report

1 major / 4 minor

Summary. The paper studies the mixed local-nonlocal quasilinear singular problem -Δ_p u + (-Δ)_q^s u = f(x)u^{-δ} in Ω, u=0 outside Ω, where f behaves like dist(x,∂Ω)^{-β}. It develops a local Hölder and gradient Hölder theory for the regular operator, proves up-to-boundary C^{1,γ} regularity under distance-like assumptions, and then applies these tools to the singular problem. The main results claimed are existence for β∈[0,p), uniqueness for β<2-1/p, boundary behavior with explicit rates depending on β+δ, optimal Sobolev regularity, non-existence for β≥p, and global Hölder regularity of the solution in Theorem 2.31 with exponents depending on β+δ. The paper is long and detailed, with many estimates following the patterns of [24] and [1].

Significance. If Theorem 2.31 were fully established, the paper would deliver a substantial result: up-to-boundary Hölder and interior gradient Hölder regularity for a doubly singular mixed local-nonlocal problem. The local regularity theorems (Theorems 2.17-2.20), the barrier constructions for boundary behavior, the comparison principle in its valid range, and the non-existence result are useful contributions that are likely to be of independent interest. The proof of the global Hölder claim, however, contains a genuine gap in the transition from local Hölder estimates to the boundary, and the uniqueness statement in Theorem 2.31 is broader than what is proved. These issues concern the central advertised results and need to be repaired before the paper can be accepted.

major comments (1)

[Theorem 2.31 vs. Theorem 2.28] Theorem 2.31 is stated for all β∈[0,p) and refers to 'the unique solution' of problem (2.13), but uniqueness is proved in Theorem 2.28 only under the additional assumption β<2-1/p. The restriction enters in Section 7.1, equation (7.39), where Hardy's inequality is used to show that the operator J_m is well-defined; this requires (1-β)p/(p-1)>-1, i.e. β<2-1/p. Without this condition, no comparison principle is established. Therefore the phrase 'the unique solution' in Theorem 2.31 is not justified for β∈[2-1/p,p). The theorem should either be restricted to the range where uniqueness is proved or the uniqueness statement should be reformulated as an open problem for the remaining range.

minor comments (4)

[Theorem 2.31(ii)] The theorem states 'u∈C^{0,η}(Ω) for all σ∈(0,1)', but the exponent should be η; the symbol σ is inconsistent with the notation η used in the same sentence.
[Section 1, Introduction] Line: 'we aim to we study Hölder regularity results' contains a typo; it should read 'we aim to study'.
[Section 7.4, Case (I)] The sentence 'set 64R=d(x)' is ambiguous; it should read 'set R=d(x)/64' to make clear that B_R(x) is a ball of radius R contained in Ω.
[Abstract and Introduction] The abstract says regularity is obtained 'albeit with different exponents depending on β+δ', but it does not mention that the uniqueness theorem is conditional on β<2-1/p; this limitation should be stated in the abstract or at least in the introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed regularity and existence results are derived from structural hypotheses and independent prior regularity theory, not from their own conclusions.

full rationale

The derivation chain is not circular. Sections 3–6 establish local and boundary regularity for the regular problem using independent results of De Filippis–Mingione [24], Antonini–Cozzi [1], Giacomoni–Kumar–Sreenadh [35], and Arora–Giacomoni–Warnault [2]; these works are not authored by the present authors, and the paper verifies their hypotheses rather than assuming the desired conclusion. The singular problem is then treated by approximation: Theorem 2.27 produces a solution in the conical shell via barrier functions and comparison, Theorem 2.28 proves uniqueness via a genuine comparison-principle argument requiring the Hardy condition β < 2 − 1/p, and Theorem 2.31 combines the boundary pointwise bound with local Hölder regularity. The conical shell bounds are pointwise estimates, not Hölder seminorms, so the claimed Hölder regularity does not reduce by definition to the boundary behavior. The only self-citation, [4], is contextual and not load-bearing. The skeptical concern about the proof of Theorem 2.31 in §7.4 — that the local Hölder constants from Theorem 2.19(a) may blow up as R → 0 when β + δ > 1 — is a potential correctness gap, not a circularity; nothing in that step is fitted, renamed, or justified solely by a self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical or mathematical entities. It relies on the stated structural assumptions on the domain, the source term, and the operator, together with standard function-space tools and cited regularity theorems.

assumptions (5)

domain assumption Omega is a bounded C^2 domain and the distance function d is C^2 in a neighborhood of the boundary.
Used to construct C^2 barriers and apply the Hopf lemma; stated in Section 2 and used throughout Section 7.
domain assumption f satisfies the two-sided bound c1 d(x)^{-beta} <= f(x) <= c2 d(x)^{-beta} in Omega_rho, with beta in [0,p).
This comparability, equation (2.14), drives the boundary behavior exponents, the existence threshold, and the non-existence theorem.
domain assumption The kernel B and monotone function Phi satisfy the structural conditions (2.6)-(2.9); for the model operator B=1 and Phi(t)=|t|^{q-2}t.
Needed for local boundedness, Hölder estimates, and the comparison principle for the general operator L.
standard math The regularity results of De Filippis-Mingione [24] and the boundary regularity of Antonini-Cozzi [1] are taken as valid.
Several key estimates are cited from these works rather than reproved, for example Lemma 4.3 and the boundary arguments in Section 6.
standard math The Hardy inequality and fractional Sobolev embeddings are applied under the stated parameter restrictions.
Used in Lemma 7.2, Theorem 2.29, and Theorem 2.30; the uniqueness argument in Section 7.1 specifically requires beta < 2 - 1/p.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Regularity results for a class of mixed local and nonlocal singular problems involving distance function." pith.science (2026). https://pith.science/paper/IWTZHPB7

@misc{pith2026241114217,
  author       = {Pith},
  title        = {Pith review of: Regularity results for a class of mixed local and nonlocal singular problems involving distance function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWTZHPB7}},
  note         = {Machine review of arXiv:2411.14217}
}

abstract

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray*} where, \begin{equation*} (-\Delta )_q^s u(x):= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} with $\Omega$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1<q\leq p<\infty$, $s\in(0,1)$, $\delta>0$ and $f\in L^\infty_{\mathrm{loc}}(\Omega)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial \Omega)^{-\beta}}$, $\beta\geq 0$ near $\partial \Omega$. We start by proving several H\"older and gradient H\"older regularity results for a more general class of quasilinear operators when $\delta=0$. Using the regularity results we deduce existence, uniqueness and H\"older regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(\Omega)$ and its behavior near $\partial \Omega$ albeit with different exponents depending on $\beta+\delta$. Boundedness and H\"older regularity result to the singular equation with critical exponent were also discussed.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators
math.AP 2026-04 conditional novelty 6.0 of 10

Weak energy solutions of a doubly nonlinear mixed local–nonlocal parabolic problem exist locally, are unique via a new Díaz–Saa inequality, extend globally by barriers, and converge to a nontrivial stationary state.
Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications
math.AP 2024-11 conditional novelty 6.0 of 10

Under p > sq, weak solutions of the mixed local-nonlocal problem with singular data are C^{1,α} up to the boundary in the weakly singular case and C^α in the strongly singular case.

Reference graph

Works this paper leans on

46 extracted references · 38 canonical work pages · cited by 2 Pith papers

[24]

Gradient regularity in mixed local and nonlocal problems

Cristiana De Filippis and Giuseppe Mingione. Gradient regularity in mixed local and nonlocal problems. Mathematische Annalen , pages 1–68, 2022

work page 2022
[1]

Global gradien t regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

Carlo Alberto Antonini and Matteo Cozzi. Global gradien t regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type. arXiv preprint arXiv:2308.06075 , 2023

arXiv 2023
[2]

Regularity results for a class of nonlinear fractional Laplacian and singular problems

Rakesh Arora, Jacques Giacomoni, and Guillaume Warnaul t. Regularity results for a class of nonlinear fractional Laplacian and singular problems. NoDEA Nonlinear Diﬀerential Equations Appl. , 28(3):Paper No. 30, 35, 2021

work page 2021
[3]

Rădulescu

Rakesh Arora and Vicenţiu D. Rădulescu. Combined eﬀects in mixed local–nonlocal stationary problems. Proceedings of the Royal Society of Edinburgh Section A: Mat hematics, page 1–47, 2023

work page 2023
[4]

Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

Kaushik Bal and Stuti Das. Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity. arXiv preprint arXiv:2405.05832 , 2024

arXiv 2024
[5]

Semilinear problems for the fractional Laplacian with a singular nonlinearity

Begoña Barrios, Ida De Bonis, María Medina, and Ireneo Pe ral. Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math., 13(1):390–407, 2015

work page 2015
[6]

A Brezis-Nirenberg type result for mixed local and nonlocal operators

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. A Brezis-Nirenberg type result for mixed local and nonlocal operators. arXiv preprint arXiv:2209.07502 , 2022

arXiv 2022
[7]

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. Mixed local and nonlocal elliptic operators: regularity and maximum principles. Comm. Partial Diﬀerential Equations , 47(3):585–629, 2022

work page 2022

Show all 46 references

[8]

A Faber-Krahn inequality for mixed local and nonlocal operators

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. A Faber-Krahn inequality for mixed local and nonlocal operators. J. Anal. Math. , 150(2):405–448, 2023

2023
[9]

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators. Math. Eng. , 5(1):Paper No. 014, 25, 2023

2023
[10]

Nec essary condition in a Brezis-Oswald-type problem for mixed local and nonlocal operators

Stefano Biagi, Dimitri Mugnai, and Eugenio Vecchi. Nec essary condition in a Brezis-Oswald-type problem for mixed local and nonlocal operators. Appl. Math. Lett. , 132:Paper No. 108177, 9, 2022

2022
[11]

A Br ezis-Oswald approach for mixed local and nonlocal operators

Stefano Biagi, Dimitri Mugnai, and Eugenio Vecchi. A Br ezis-Oswald approach for mixed local and nonlocal operators. Commun. Contemp. Math. , 26(2):Paper No. 2250057, 28, 2024

2024
[12]

Multiplicity of posi tive solutions for mixed local-nonlocal singular critical problems

Stefano Biagi and Eugenio Vecchi. Multiplicity of posi tive solutions for mixed local-nonlocal singular critical problems. Calc. Var. Partial Diﬀerential Equations , 63(9):Paper No. 221, 45, 2024

2024
[13]

Semilinear elliptic equations involving mixed local and nonlocal operators

Stefano Biagi, Eugenio Vecchi, Serena Dipierro, and En rico Valdinoci. Semilinear elliptic equations involving mixed local and nonlocal operators. Proc. Roy. Soc. Edinburgh Sect. A , 151(5):1611–1641, 2021

2021
[14]

Semilinear elliptic e quations with singular nonlinearities

Lucio Boccardo and Luigi Orsina. Semilinear elliptic e quations with singular nonlinearities. Calc. Var. Partial Diﬀerential Equations , 37(3-4):363–380, 2010. 46

2010
[15]

On the Hö lder regularity of signed solutions to a doubly nonlinear equation

Verena Bögelein, Frank Duzaar, and Naian Liao. On the Hö lder regularity of signed solutions to a doubly nonlinear equation. J. Funct. Anal. , 281(9):Paper No. 109173, 58, 2021

2021
[16]

The second eigenvalue o f the fractional p-Laplacian

Lorenzo Brasco and Enea Parini. The second eigenvalue o f the fractional p-Laplacian. Adv. Calc. Var. , 9(4):323–355, 2016

2016
[17]

Functional analysis, Sobolev spaces and partial diﬀerentia l equations, volume 2

Haim Brézis. Functional analysis, Sobolev spaces and partial diﬀerentia l equations, volume 2. Springer, 2011

2011
[18]

A system of local-nonlocal p- Laplacians: the eigenvalue problem and its asymptotic limi t asp → ∞

Stefano Buccheri, João Vítor da Silva, and Luís Henriqu e de Miranda. A system of local-nonlocal p- Laplacians: the eigenvalue problem and its asymptotic limi t asp → ∞. Asymptotic Analysis, 128(2):149–181, 2022

2022
[19]

Nonlocal problems with singular nonlinearity

Annamaria Canino, Luigi Montoro, Berardino Sciunzi, a nd Marco Squassina. Nonlocal problems with singular nonlinearity. Bull. Sci. Math. , 141(3):223–250, 2017

2017
[20]

Existence and uniqueness for p-Laplace equations involving singular nonlinearities

Annamaria Canino, Berardino Sciunzi, and Alessandro T rombetta. Existence and uniqueness for p-Laplace equations involving singular nonlinearities. NoDEA Nonlinear Diﬀerential Equations Appl. , 23(2):Art. 8, 18, 2016

2016
[21]

Linear and nonlinear functional analysis with application s

Philippe G Ciarlet. Linear and nonlinear functional analysis with application s. SIAM, 2013

2013
[22]

M. G. Crandall, P. H. Rabinowitz, and L. Tartar. On a Diri chlet problem with a singular nonlinearity. Comm. Partial Diﬀerential Equations , 2(2):193–222, 1977

1977
[23]

Comparison theorems for some quasil inear degenerate elliptic operators and applications to symmetry and monotonicity results

Lucio Damascelli. Comparison theorems for some quasil inear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré C Anal. Non Linéaire , 15(4):493–516, 1998

1998
[25]

Inter ior and boundary regularity of mixed local nonlocal problem with singular data and its applications

R Dhanya, Jacques Giacomoni, and Ritabrata Jana. Inter ior and boundary regularity of mixed local nonlocal problem with singular data and its applications. arXiv preprint arXiv:2411.18505 , 2024

2024 arXiv
[26]

Hitchhiker’s guide to the fractional Sobolev spaces

Eleonora Di Nezza, Giampiero Palatucci, and Enrico Val dinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin des sciences mathématiques , 136(5):521–573, 2012

2012
[27]

Degenerate parabolic equations

Emmanuele DiBenedetto. Degenerate parabolic equations. Springer Science & Business Media, 2012

2012
[28]

Partial Diﬀerential Equations , volume 19

Lawrence C Evans. Partial Diﬀerential Equations , volume 19. American Mathematical Soc., 2010

2010
[29]

On a degenerate singular elliptic pr oblem

Prashanta Garain. On a degenerate singular elliptic pr oblem. Math. Nachr. , 295(7):1354–1377, 2022

2022
[30]

On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlin- earity

Prashanta Garain. On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlin- earity. J. Geom. Anal. , 33(7):Paper No. 212, 20, 2023

2023
[31]

On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular proble ms

Prashanta Garain, Wontae Kim, and Juha Kinnunen. On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular proble ms. Forum Math., 36(3):697–715, 2024

2024
[32]

On the regularity t heory for mixed local and nonlocal quasilinear elliptic equations

Prashanta Garain and Juha Kinnunen. On the regularity t heory for mixed local and nonlocal quasilinear elliptic equations. Trans. Amer. Math. Soc. , 375(8):5393–5423, 2022

2022
[33]

Higher Hölder regu larity for mixed local and nonlocal degenerate elliptic equations

Prashanta Garain and Erik Lindgren. Higher Hölder regu larity for mixed local and nonlocal degenerate elliptic equations. Calculus of Variations and Partial Diﬀerential Equations , 62(2):67, 2023

2023
[34]

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

Prashanta Garain and Alexander Ukhlov. Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems. Nonlinear Anal., 223:Paper No. 113022, 35, 2022

2022
[35]

Interior and boundary regularity results for strongly nonhomogeneous p,q -fractional problems

Jacques Giacomoni, Deepak Kumar, and Konijeti Sreenad h. Interior and boundary regularity results for strongly nonhomogeneous p,q -fractional problems. Adv. Calc. Var. , 16(2):467–501, 2023

2023
[36]

Sob olev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, and Peter Takáč. Sob olev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 6(1):117–158, 2007. 47

2007
[37]

Elliptic partial diﬀerential equations of second order , volume 224

David Gilbarg and Neil S Trudinger. Elliptic partial diﬀerential equations of second order , volume 224. Springer
[38]

Quasilinear ellipt ic equations involving critical Sobolev exponents

Mohammed Guedda and Laurent Véron. Quasilinear ellipt ic equations involving critical Sobolev exponents. Nonlinear Anal., 13(8):879–902, 1989

1989
[39]

Multiplicity and asymptotic behavior of p ositive solutions for a singular semilinear elliptic problem

Yang Haitao. Multiplicity and asymptotic behavior of p ositive solutions for a singular semilinear elliptic problem. J. Diﬀerential Equations , 189(2):487–512, 2003

2003
[40]

A. C. Lazer and P. J. McKenna. On a singular nonlinear ell iptic boundary-value problem. Proc. Amer. Math. Soc. , 111(3):721–730, 1991

1991
[41]

Lieberman

Gary M. Lieberman. Boundary regularity for solutions o f degenerate elliptic equations. Nonlinear Anal. , 12(11):1203–1219, 1988

1988
[42]

Fractional eigenva lues

Erik Lindgren and Peter Lindqvist. Fractional eigenva lues. Calc. Var. Partial Diﬀerential Equations , 49(1- 2):795–826, 2014

2014
[43]

Fine regularity of soluti ons of elliptic partial diﬀerential equations

Jan Mal` y and William Ziemer. Fine regularity of soluti ons of elliptic partial diﬀerential equations. Mathe- matical Surveys and Monographs , 1997

1997
[44]

Regularity of the gradient for a class of nonlinear possibly d egenerate elliptic equations

Juan Jose Manfredi. Regularity of the gradient for a class of nonlinear possibly d egenerate elliptic equations . Washington University in St. Louis, 1986

1986
[45]

Regularity results for solutions of mixed local and nonlocal elliptic equations

Xifeng Su, Enrico Valdinoci, Yuanhong Wei, and Jiwen Zh ang. Regularity results for solutions of mixed local and nonlocal elliptic equations. Math. Z. , 302(3):1855–1878, 2022

2022
[46]

On some regularity properties of mixed local and nonlocal elliptic equations

Xifeng Su, Enrico Valdinoci, Yuanhong Wei, and Jiwen Zh ang. On some regularity properties of mixed local and nonlocal elliptic equations. J. Diﬀerential Equations , 416:576–613, 2025. 48

2025

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [24]

Gradient regularity in mixed local and nonlocal problems

Cristiana De Filippis and Giuseppe Mingione. Gradient regularity in mixed local and nonlocal problems. Mathematische Annalen , pages 1–68, 2022

work page 2022

[2] [1]

Global gradien t regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

Carlo Alberto Antonini and Matteo Cozzi. Global gradien t regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type. arXiv preprint arXiv:2308.06075 , 2023

arXiv 2023

[3] [2]

Regularity results for a class of nonlinear fractional Laplacian and singular problems

Rakesh Arora, Jacques Giacomoni, and Guillaume Warnaul t. Regularity results for a class of nonlinear fractional Laplacian and singular problems. NoDEA Nonlinear Diﬀerential Equations Appl. , 28(3):Paper No. 30, 35, 2021

work page 2021

[4] [3]

Rădulescu

Rakesh Arora and Vicenţiu D. Rădulescu. Combined eﬀects in mixed local–nonlocal stationary problems. Proceedings of the Royal Society of Edinburgh Section A: Mat hematics, page 1–47, 2023

work page 2023

[5] [4]

Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

Kaushik Bal and Stuti Das. Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity. arXiv preprint arXiv:2405.05832 , 2024

arXiv 2024

[6] [5]

Semilinear problems for the fractional Laplacian with a singular nonlinearity

Begoña Barrios, Ida De Bonis, María Medina, and Ireneo Pe ral. Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math., 13(1):390–407, 2015

work page 2015

[7] [6]

A Brezis-Nirenberg type result for mixed local and nonlocal operators

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. A Brezis-Nirenberg type result for mixed local and nonlocal operators. arXiv preprint arXiv:2209.07502 , 2022

arXiv 2022

[8] [7]

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, and E ugenio Vecchi. Mixed local and nonlocal elliptic operators: regularity and maximum principles. Comm. Partial Diﬀerential Equations , 47(3):585–629, 2022

work page 2022