Pith. sign in

REVIEW 2 major objections 4 minor 44 references

Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A mixed local-nonlocal singular PDE admits weak solutions when the singular source is a measure, including the purely singular case.

desk verdict A credible new existence result for singular measure data that currently overclaims because Lemma 4.1 needs an inf_Ω δ > 0 hypothesis not stated in Theorem 2.8. read the letter →

arxiv 2507.04260 v1 pith:DVVF3S6Z submitted 2025-07-06 math.AP

Sanjit Biswas , Prashanta Garain This is my paper · ORCID
classification math.AP MSC 35M1035M1235J7535R0635R11
keywords mixedlocal-nonlocaloperatorvariablesingularexponentmeasuredataweaksolutionp-capacityweak-Lpspacesexistence
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 establishes existence of weak solutions to the semilinear mixed local-nonlocal problem $\mathcal{M}u = \nu/u^{\delta(x)} + \mu$ in a bounded Lipschitz domain, with zero exterior condition and positivity inside the domain. The main result, Theorem 2.8, allows both source terms $\mu$ and $\nu$ to be non-negative bounded Radon measures, with $\nu$ belonging to the class $M^p_0(\Omega)$ of measures absolutely continuous with respect to $p$-capacity, for $1

What carries the argument

The argument is carried by monotone approximation of the singular measure $\nu$ by measures $\nu_n$ in $W^{-1,p'}(\Omega)$, combined with the decomposition $\nu=H-\operatorname{div}G$ available for measures in $M^p_0(\Omega)$. Each approximating problem shifts the singularity to $1/(u_n+1/n)^{\delta(x)}$, solves the resulting equation in $W^{1,2}_0(\Omega)$ through the standard coercive-bilinear-form theorem after a further regularization, and then passes to the limit. Uniform positivity on compact subsets is obtained by comparing every $u_n$ with a fixed positive solution attached to the first approximating measure. The limiting passage uses the cutoff functions $T_k(s)=\max\{-k,\min\{s,k\}\}$ and tails $G_k(s)=(|s|-k)_+\operatorname{sgn}(s)$ to obtain uniform bounds in weak-$L^q$ spaces, which yield $W^{1,p}_{\mathrm{loc}}$ convergence and the stated regularity of truncations.

What would settle it

Take a bounded Lipschitz domain and $\delta(x)=x_1$ on a domain touching the hyperplane $x_1=0$; this satisfies every hypothesis of Theorem 2.8 except any hidden positive lower bound, because $\inf_\Omega\delta=0$. At estimate (4.6) the proof controls $\log(|u_n|+\tau)/(|u_n|+\tau)^{\delta(x)}$ using $r=\inf_\Omega\delta>0$, which is absent here, so checking whether the approximate problems of Lemma 4.1 still admit the asserted uniform bounds for this $\delta$ would settle whether the stated hypotheses are sufficient.

Watch

Extended reading notes

Core claim

The central claim is that a continuous, locally Lipschitz variable exponent $\delta:\Omega\to(0,\infty)$ with $|\nabla\delta|\in L^N(\Omega)$ and a boundary-side upper bound, together with non-negative bounded Radon measures $\mu$ and $\nu\ne0$ with $\nu\in M^p_0(\Omega)$ and $1<p<N/(N-1)$, guarantees a weak solution $u\in W^{1,p}_{\mathrm{loc}}(\Omega)\cap L^1(\Omega)$ of problem (1.1) in the sense of Definition 2.4. When the boundary-side bound is $\delta_*=1$, the solution lies in $W^{1,p}_0(\Omega)$; for $\delta_*>1$, the truncated powers $T_k^{\frac{\delta_*+1}{2}}(u)$ lie in $W^{1,2}_0(\Omega)$ for every $k>0$. A second theorem for constant $\delta$ gives existence for $\nu\in M^p_0(\Omega)$ with exponent $p=N(\delta+1)/(N+\delta-1)$ when $0<\delta<1$ and $p=2$ when $\delta\ge1$, assuming $\mu\in L^{N(\delta+1)/(N+2\delta)}(\Omega)$. The paper identifies the purely singular $\nu$ case as the main novelty.

Load-bearing premise

The proof needs the singular exponent to stay bounded away from zero everywhere in the domain, but the theorem only assumes it is positive, continuous, locally Lipschitz, and with gradient in $L^N$; nothing in those assumptions prevents $\delta$ from tending to zero at the boundary.

Editorial extensions

If this is right

  • If Theorem 2.8 is correct, singular measure sources are admissible in mixed local-nonlocal singular equations, a class previously limited to absolutely continuous or integrable data.
  • For boundary-side exponent bound $\delta_*=1$, the constructed solution belongs to $W^{1,p}_0(\Omega)$; for $\delta_*>1$, the solution lies in $W^{1,p}_{\mathrm{loc}}(\Omega)\cap L^1(\Omega)$ with truncated powers in $W^{1,2}_0(\Omega)$.
  • Theorem 2.10 gives the analogous existence result for constant $\delta$, with $\mu$ only required to lie in $L^{N(\delta+1)/(N+2\delta)}(\Omega)$ and with $p$ depending on $\delta$.
  • By Remark 2.11, the proof also covers the purely local operator $\operatorname{div}(A(x)\nabla u)$, so the singular-measure existence result is new even in the purely local setting.
  • When $\nu$ is non-singular, Theorem 2.8 recovers the earlier variable-exponent existence theorem under the added assumption $|\nabla\delta|\in L^N(\Omega)$.

Reading between the lines

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

  • The proof of Lemma 4.1 appears to require an unstated global lower bound $\inf_\Omega\delta>0$; since the hypotheses only say $\delta>0$ pointwise, the stated argument leaves open the case where $\delta$ tends to zero at the boundary.
  • The monotone-measure strategy is not tied to the particular mixed operator, so the same decomposition and truncation estimates should transfer to quasilinear or anisotropic local-nonlocal operators once the corresponding capacity class of measures is fixed.
  • A natural testable extension is a solution-dependent singular exponent $\delta(x,u)$, since the machinery already handles spatial dependence together with measure data; the paper does not address that case.
  • The existence result invites finer questions about summability and regularity of solutions when $\nu$ is singular, since the paper establishes existence and the stated truncation regularity but not optimal integrability.
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.

Referee Report

2 major / 4 minor

Summary. The paper studies the mixed local-nonlocal semilinear problem (1.1) with a variable singular exponent δ(x) and with both source terms allowed to be non-negative bounded Radon measures. The main result, Theorem 2.8, claims existence of a weak solution in W^{1,p}_{loc}(Ω) ∩ L^1(Ω) whenever ν ∈ M^p_0(Ω) is nonzero with 1 < p < N/(N-1), under regularity conditions on δ. A second result, Theorem 2.10, treats the constant-exponent case with a more general class of measures. The proof proceeds by approximating ν with measures in W^{-1,p'}, solving regularized problems from Lemma 4.1, and passing to the limit via a priori estimates. The authors emphasize that the singular-measure case is new even for constant exponents.

Significance. If the main results were fully established, the paper would provide a meaningful extension of the available existence theory for mixed local-nonlocal singular problems: the treatment of purely singular measure data with variable singular exponents appears not to be contained in the cited literature, and the paper builds on the approximation scheme of [41] as well as on the authors' earlier work in [15]. The functional setting with p-capacity measures is appropriate, and the paper is careful to identify which estimates are new. However, two gaps in the written proofs — one in the construction of the approximate solutions and one in the proof of Theorem 2.10 — prevent the stated theorems from being considered proven in their full generality at this stage.

major comments (2)
  1. [§4.1, Lemma 4.1, estimate (4.6)] The proof of Lemma 4.1 relies on the uniform bound |log(|u_n|+τ)/(|u_n|+τ)^{δ(x)}| ≤ C in (4.6), justified by setting r := inf_Ω δ > 0. This global lower bound is not a hypothesis of Theorem 2.8: the hypotheses only assume δ is continuous, locally Lipschitz, |∇δ| ∈ L^N(Ω), and δ satisfies P_{ε,δ*}, all of which are compatible with δ(x) = x_1 on a ball, where inf_Ω δ = 0. Without inf_Ω δ > 0, the logarithmic factor is not uniformly bounded with respect to the spatial variable and the size of u_n, so estimate (4.6) fails and the uniform W^{1,2}_0 bound on {u_n} obtained from (4.4)-(4.5) is not established. Since Lemma 4.1 is used to build the approximate solutions for both Theorem 2.8 and Theorem 2.10, this is a load-bearing gap. The issue is fixable by adding an explicit assumption δ ≥ δ_0 > 0 in Ω, but as written Theorem 2.8 overclaims.
  2. [§3.2, proof of Theorem 2.10] The proof of Theorem 2.10 begins by imposing an additional condition not present in the theorem statement: it says, in §3.2, that ν is assumed to be singular with respect to the Lebesgue measure. The statement of Theorem 2.10 allows an arbitrary non-negative bounded Radon measure ν ∈ M^p_0(Ω) \ {0}. The limiting procedure for the singular measure term is carried out only under this extra restriction, and the passage for a nontrivial absolutely continuous part of ν is not addressed. Consequently, the proof does not establish the theorem as stated. The authors should either add the singularity assumption to the theorem statement (and adjust the abstract/introduction accordingly) or explicitly invoke the non-singular case from [15] in the proof of Theorem 2.10.
minor comments (4)
  1. [§3.1, estimate (3.6)] The notation ||C^{δ(x)}||_{L^∞(Ω)} is confusing; the constant from the lower bound on u_n should be denoted more clearly, e.g., by a constant depending on C(ω) and sup_{ω} δ.
  2. [§4.1, equation (4.7)] There is a typographical issue in the displayed chain: the right-hand side appears twice with an intervening line break and a misplaced equal sign; the equation should be written as a single alignment.
  3. [§4.3, Lemma 4.4] The statement of Lemma 4.4 introduces a function g, while Theorem 2.10 uses µ. Align the notation, e.g., by defining g = µ or by replacing g with µ throughout the lemma.
  4. [§1, Introduction] The sentence 'with the singular component modeled by both a singular and non-singular measure' is ambiguous: it could be read as saying the measure itself is singular, whereas the intended meaning is that the nonlinearity 1/u^{δ(x)} is singular. Consider rewording.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation is apparent: the singular-measure existence claim is obtained by regularization and a limiting argument, not by assuming the conclusion. Heavy use of the authors' earlier paper [15] is methodological; the only flagged gap (inf_Ω δ > 0 used but not assumed) is a correctness/premise issue, not circularity.

full rationale

The central claim (Theorem 2.8), existence of weak solutions when the measure ν is purely singular, is not an input to the proof. The proof regularizes ν via a monotone approximation ν_n (Lemma 2.5), solves the approximate problems (3.1), and then passes to the limit using the a priori estimates of Lemma 4.3. The limiting process is not assumed to produce the target solution; it is derived. The paper does cite the authors' own prior work [15] for structural estimates and for the approximate construction ("As in [15, Lemma A.1], there exists u_n ∈ W_0^{1,2}(Ω) satisfying the equation (4.3)" and "along the lines of proof of the estimate of (3.12) on [15, page 14]"). This is heavy self-citation, but [15] is an earlier independent result for non-singular measure data; the purely singular case is explicitly distinguished as new and is not contained in the cited statements. Therefore the self-citation is not a self-contained loop that reduces Theorem 2.8 to its own conclusion. The one substantive red flag is non-circular: in Lemma 4.1, estimate (4.6) is justified by "where r := inf_Ω δ > 0", but Theorem 2.8 only assumes δ is continuous, locally Lipschitz, |∇δ| ∈ L^N(Ω), and satisfies P_{ε,δ*}; δ may vanish at the boundary, in which case the uniform bound on log(|u_n|+τ)/(|u_n|+τ)^{δ(x)} is not available. This is a missing hypothesis in the proof as written, not a reduction of the prediction to its input. Score 2 reflects the methodological dependence on the authors' prior work and the flagged boundary-premise gap, not a circular derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented entities. The derivation rests on standard functional analysis plus one unstated lower-bound assumption on delta and several cited results, including prior work by the authors, used without full proof.

assumptions (6)
  • domain assumption Omega is a bounded Lipschitz domain in R^N with N > 2.
    Stated in the introduction; all Sobolev, fractional Sobolev, and capacity results depend on it.
  • domain assumption The coefficient matrix A is uniformly elliptic and bounded, and the kernel K satisfies the symmetric bounds (1.2) and (1.3).
    These define the mixed local-nonlocal operator M and are used throughout the proofs.
  • standard math Standard Sobolev, fractional Sobolev, Marcinkiewicz, measure decomposition, and p-capacity results from the literature hold as cited.
    Invoked throughout Section 2 and the appendix for embeddings, capacity characterizations, and measure approximations.
  • standard math For every nonnegative measure in M^p_0, there is an increasing sequence of W^{-1,p'} measures converging in total variation (Lemma 2.5).
    Cited to [8,23]; this approximation underpins the whole proof of Theorem 2.8.
  • standard math The cited positivity and regularity result [31, Lemma 4.6] applies to the mixed local-nonlocal operator with singular right-hand side.
    Used in Lemma 4.2 to conclude that the comparison solution w is strictly positive on compact subsets, giving uniform positivity of the approximating solutions.
  • ad hoc to paper The variable exponent delta has a positive global infimum r = inf_Omega delta > 0.
    Not stated in the theorems, but used in Lemma 4.1, equation (4.6), to bound log terms. Positive continuous functions on open bounded domains need not satisfy this, so the proofs require an extra hypothesis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures." pith.science (2026). https://pith.science/paper/DVVF3S6Z

@misc{pith2026250704260,
  author       = {Pith},
  title        = {Pith review of: Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVVF3S6Z}},
  note         = {Machine review of arXiv:2507.04260}
}
read the original abstract

This article establishes the existence of weak solutions for a class of mixed local-nonlocal problems with pure and perturbed singular nonlinearities. A key novelty is the treatment of variable singular exponents alongside measure-valued data. Notably, both source terms may be measures, with the singular component modeled by both a singular and non-singular measure. Our main focus is on the singular measure data, which appears to be new, even for constant exponents.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [15]

    Regularity and existence for semilinear mixed local–nonlocal equations with variable singularities and measure data

    Sanjit Biswas and Prashanta Garain. Regularity and existence for semilinear mixed local–nonlocal equations with variable singularities and measure data. Communications in Contemporary Mathematics , 0(0):2550028, 0

    work page

  2. [41]

    A Lazer-McKenna type problem with measures

    Luigi Orsina and Francesco Petitta. A Lazer-McKenna type problem with measures. Differential Integral Equations, 29(1-2):19–36, 2016

    work page 2016

  3. [1]

    Alves, Carlos Alberto Santos, and Thiago Willians Siqueira

    Claudianor O. Alves, Carlos Alberto Santos, and Thiago Willians Siqueira. Uniqueness in W 1,p(x) loc (Ω) and continuity up to portions of the boundary of positive solutions for a strongly-singular elliptic problem. J. Differential Equations , 269(12):11279–11327, 2020

    work page 2020

  4. [2]

    Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization, 2024 (to appear in Math Z)

    Gurdev Chand Anthal and Prashanta Garain. Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization, 2024 (to appear in Math Z)

    work page 2024

  5. [3]

    Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities

    David Arcoya and Lucio Boccardo. Multiplicity of solutions for a Dirichlet problem with a singular and a supercritical nonlinearities. Differential Integral Equations, 26(1-2):119– 128, 2013

    work page 2013

  6. [4]

    Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity

    David Arcoya and Lourdes Moreno-M´ erida. Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity. Nonlinear Anal., 95:281–291, 2014

    work page 2014

  7. [5]

    Radulescu

    Rakesh Arora and Vicentiu D. Radulescu. Combined effects in mixed local-nonlocal sta- tionary problems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. Published online 2023:1-47 , 2023

    work page 2023

  8. [6]

    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 e-prints , page arXiv:2405.05832, May 2024

    work page Pith review arXiv 2024

Show all 44 references
  1. [7]

    On an anisotropic p-Laplace equation with variable singular exponent

    Kaushik Bal, Prashanta Garain, and Tuhina Mukherjee. On an anisotropic p-Laplace equation with variable singular exponent. Adv. Differential Equations , 26(11-12):535– 562, 2021

    2021

  2. [8]

    Baras and M

    P. Baras and M. Pierre. Singularit´ es ´ eliminables pour des ´ equations semi-lin´ eaires.Ann. Inst. Fourier (Grenoble), 34(1):185–206, 1984

    1984

  3. [9]

    Existence results for mixed local and nonlo- cal elliptic equations involving singularity and nonregular data

    Souvik Bhowmick and Sekhar Ghosh. Existence results for mixed local and nonlo- cal elliptic equations involving singularity and nonregular data. arXiv e-prints , page arXiv:2410.04441, October 2024

    2024 arXiv

  4. [10]

    A Hong-Krahn- Szeg¨ o inequality for mixed local and nonlocal operators.Math

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

    2023

  5. [11]

    A Brezis-Oswald approach for mixed local and nonlocal operators

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

    2024

  6. [12]

    Multiplicity of positive solutions for mixed local- nonlocal singular critical problems

    Stefano Biagi and Eugenio Vecchi. Multiplicity of positive solutions for mixed local- nonlocal singular critical problems. Calc. Var. Partial Differential Equations, 63(9):Paper No. 221, 45, 2024. Mixed local-nonlocal singular problem 23

    2024

  7. [13]

    Mixed local and nonlocal equation with singular nonlinearity having variable exponent

    Kheireddine Biroud. Mixed local and nonlocal equation with singular nonlinearity having variable exponent. J. Pseudo-Differ. Oper. Appl. , 14(1):Paper No. 13, 24, 2023

    2023

  8. [14]

    Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R2, 2025

    Sanjit Biswas. Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R2, 2025

    2025

  9. [16]

    Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

    Lucio Boccardo, Thierry Gallou¨ et, and Luigi Orsina. Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 13(5):539–551, 1996

    1996

  10. [17]

    Semilinear elliptic equations with singular nonlinear- ities

    Lucio Boccardo and Luigi Orsina. Semilinear elliptic equations with singular nonlinear- ities. Calc. Var. Partial Differential Equations , 37(3-4):363–380, 2010

    2010

  11. [18]

    Buccheri, J

    S. Buccheri, J. V. da Silva, and L. H. de Miranda. A system of local/nonlocalp-laplacians: the eigenvalue problem and its asymptotic limit asp → ∞. Asymptotic Anal., 128(2):149– 181, 2022

    2022

  12. [19]

    Nonlocal problems with singular nonlinearity

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

    2017

  13. [20]

    Mart ´ ınez-Aparicio

    Jos´ e Carmona and Pedro J. Mart ´ ınez-Aparicio. A singular semilinear elliptic equation with a variable exponent. Adv. Nonlinear Stud. , 16(3):491–498, 2016

    2016

  14. [21]

    New techniques for solving some class of singular elliptic equations

    Michel Chipot and Linda Maria De Cave. New techniques for solving some class of singular elliptic equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 29(3):487– 510, 2018

    2018

  15. [22]

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

    1977

  16. [23]

    On the integral representation of certain local functionals

    Gianni Dal Maso. On the integral representation of certain local functionals. Ricerche Mat., 32(1):85–113, 1983

    1983

  17. [24]

    Renormalized solutions of elliptic equations with general measure data

    Gianni Dal Maso, Fran¸cois Murat, Luigi Orsina, and Alain Prignet. Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 28(4):741–808, 1999

    1999

  18. [25]

    Hitchhiker’s guide to the fractional Sobolev spaces

    Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. , 136(5):521–573, 2012

    2012

  19. [26]

    Partial differential equations , volume 19

    Lawrence C Evans. Partial differential equations , volume 19. American Mathematical Society, 2022. Mixed local-nonlocal singular problem 24

    2022

  20. [27]

    Existence, Uniqueness of Positive Solution to a Fractional Laplacians with Singular Nonlinearity

    Yanqin Fang. Existence, Uniqueness of Positive Solution to a Fractional Laplacians with Singular Nonlinearity. arXiv e-prints , page arXiv:1403.3149, March 2014

    2014 arXiv

  21. [28]

    On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity

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

    2023

  22. [29]

    Mixed anisotropic and nonlocal Sobolev type inequalities with ex- tremal

    Prashanta Garain. Mixed anisotropic and nonlocal Sobolev type inequalities with ex- tremal. J. Math. Sci., New York , 281(5):633–645, 2024

    2024

  23. [30]

    On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem

    Prashanta Garain. On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem. Manuscripta Math. , 174(1-2):141–158, 2024

    2024

  24. [31]

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

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

    2024

  25. [32]

    Quasilinear nonlocal elliptic problems with variable singular exponent

    Prashanta Garain and Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Commun. Pure Appl. Anal. , 19(11):5059–5075, 2020

    2020

  26. [33]

    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

  27. [34]

    Sekhar Ghosh, Debajyoti Choudhuri, and Ratan Kr. Giri. Singular nonlocal problem involving measure data. Bull. Braz. Math. Soc. (N.S.) , 50(1):187–209, 2019

    2019

  28. [35]

    Nonlinear potential theory of de- generate elliptic equations

    Juha Heinonen, Tero Kilpel¨ ainen, and Olli Martio. Nonlinear potential theory of de- generate elliptic equations . Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original

    2006

  29. [36]

    Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities

    Norimichi Hirano, Claudio Saccon, and Naoki Shioji. Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations, 9(1-2):197–220, 2004

    2004

  30. [37]

    Lazer-mckenna type problem involving mixed local and nonlocal elliptic operators

    Shuibo Huang and Hichem Hajaiej. Lazer-mckenna type problem involving mixed local and nonlocal elliptic operators. Nonlinear Differential Equations and Applications , 03 2023

    2023

  31. [38]

    Sur l’interpolation d’op´ erations

    Joseph Marcinkiewicz. Sur l’interpolation d’op´ erations. C. R. Acad. Sci., Paris , 208:1272–1273, 1939

    1939

  32. [39]

    On singular elliptic equations with measure sources

    Francescantonio Oliva and Francesco Petitta. On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var. , 22(1):289–308, 2016

    2016

  33. [40]

    Singular elliptic pdes: an extensive overview, 2024

    Francescantonio Oliva and Francesco Petitta. Singular elliptic pdes: an extensive overview, 2024. Mixed local-nonlocal singular problem 25

    2024

  34. [42]

    Existence of solutions to semilinear elliptic equations with variable exponent singularity

    Ambesh Kumar Pandey and Rasmita Kar. Existence of solutions to semilinear elliptic equations with variable exponent singularity. Rocky Mountain J. Math. , 53(3):903–913, 2023

    2023

  35. [43]

    Real analysis

    Halsey Royden and Patrick Michael Fitzpatrick. Real analysis. China Machine Press, 2010

    2010

  36. [44]

    Salort and Eugenio Vecchi

    Ariel M. Salort and Eugenio Vecchi. On the mixed local-nonlocal H´ enon equation.Differ. Integral Equ., 35(11-12):795–818, 2022

    2022

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.