Singular semilinear elliptic equations in nondivergence form

Agnieszka Ka{\l}amajska; Artur Rutkowski; Dalimil Pe\v{s}a

arxiv: 2605.03704 · v1 · submitted 2026-05-05 · 🧮 math.AP · math.FA

Singular semilinear elliptic equations in nondivergence form

Agnieszka Ka{\l}amajska , Dalimil Pe\v{s}a , Artur Rutkowski This is my paper

Pith reviewed 2026-05-07 15:01 UTC · model grok-4.3

classification 🧮 math.AP math.FA

keywords semilinear elliptic equationsnondivergence formsingular nonlinearityexistence and uniquenessDirichlet boundary conditionsGagliardo-Nirenberg inequalitiesGreen function estimatesKato inequalities

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The pith

Solutions exist and are unique for the singular semilinear equation -Pu = f/u^γ in nondivergence form when the domain and operator coefficients meet stated smoothness thresholds.

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

The paper establishes existence of solutions to the singular semilinear elliptic equation -Pu = f/u^γ with zero Dirichlet boundary values, where P is a second-order elliptic operator written in nondivergence form. Existence holds when the bounded domain Ω belongs to the class C^{1,1} and the coefficients of P are C^1, while uniqueness of solutions inside L^1(Ω) holds when Ω belongs to C^2 and the coefficients are C^2. A sympathetic reader would care because these equations arise in models that develop strong singularities near the boundary, and guaranteed existence plus uniqueness supplies a reliable starting point for analysis and approximation. The proofs proceed by combining nonlinear variants of Gagliardo-Nirenberg inequalities, estimates for the Green function of P, and new Kato-type inequalities to control the singular term f/u^γ.

Core claim

We study the singular semilinear equation -Pu = f/u^γ on a bounded domain Ω with Dirichlet condition u ≡ 0 on ∂Ω, where P is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that Ω ∈ C^{1,1} and P has C^1 coefficients, as well as the uniqueness of solutions in L^1(Ω), under the assumptions that Ω ∈ C^2 and P has C^2 coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities.

What carries the argument

Nonlinear Gagliardo-Nirenberg inequalities together with Green-function estimates and Kato-type inequalities that tame the singular right-hand side f/u^γ for the nondivergence operator P.

If this is right

A solution in the appropriate function space exists whenever the domain is C^{1,1} and the coefficients of P are C^1.
Uniqueness inside L^1(Ω) follows once the domain and coefficients are upgraded to C^2.
The singular forcing term remains compatible with integrability of the solution under the stated regularity.
The same combination of inequalities supplies a route to treat nondivergence operators that is not available from divergence-form techniques alone.

Where Pith is reading between the lines

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

The inequality toolkit developed here may adapt directly to time-dependent versions of the same singular equation.
Uniqueness in L^1 could justify convergence of numerical schemes that approximate the solution from discrete data.
The approach may extend to systems of several such singular equations coupled through the right-hand side.

Load-bearing premise

The domain boundary must be C^{1,1} smooth and the coefficients of P must be C^1 for existence to hold, or both must be one derivative smoother for uniqueness in L^1.

What would settle it

A concrete C^{1,1} domain together with C^1 coefficients of P and suitable f and γ for which either no integrable solution exists or at least two distinct L^1 solutions can be exhibited.

read the original abstract

We study the singular semilinear equation $-Pu = \frac{f}{u^\gamma}$ on a bounded domain $\Omega$ with Dirichlet condition $u \equiv 0$ on $\partial \Omega$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $\Omega \in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(\Omega)$, under the assumptions that $\Omega \in C^2$ and $P$ has $C^2$ coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo--Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities.

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 gives existence for -Pu = f/u^γ in nondivergence form under C^{1,1} domains and C^1 coefficients, plus L^1 uniqueness under C^2 regularity, via nonlinear Gagliardo-Nirenberg, Green functions, and Kato inequalities.

read the letter

The main thing here is a clean existence result for the singular semilinear equation in nondivergence form when the domain is only C^{1,1} and the coefficients are C^1, together with uniqueness in L^1 under the stronger C^2 assumptions. They reach this by putting together nonlinear Gagliardo-Nirenberg inequalities, Green function estimates, and Kato-type inequalities, with f positive continuous and 0 < γ < 1 fixed in the usual range for these problems. That combination is the actual novelty, since nondivergence operators often need separate handling from the divergence case. The paper states the regularity thresholds explicitly, which is useful for anyone trying to apply or extend the result. The proofs appear to close without extra hidden smoothness on the data, at least according to the details that were checked. One soft spot is that uniqueness stays in L^1 rather than stronger norms, though this is standard for singular right-hand sides and does not break the argument. Another is whether the cited nonlinear inequalities apply directly to the nondivergence operator without extra work; the paper claims they do, and nothing in the setup suggests a mismatch. This is aimed at people working on singular elliptic equations, especially those who care about explicit regularity conditions rather than abstract existence. A reader in that subfield would get concrete theorems and a reusable tool mix. It deserves a serious referee because the claims are specific, the assumptions are stated up front, and the approach uses established estimates in a new setting.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the singular semilinear elliptic problem −Pu = f/u^γ on a bounded domain Ω with zero Dirichlet boundary data, where P is a second-order elliptic operator in nondivergence form. It establishes existence of a solution when Ω ∈ C^{1,1} and the coefficients of P are C^1, together with uniqueness in L^1(Ω) when Ω ∈ C^2 and the coefficients are C^2. The proofs combine nonlinear Gagliardo–Nirenberg inequalities, Green-function estimates, and new variants of Kato-type inequalities, with the full text specifying f > 0 continuous and 0 < γ < 1.

Significance. If the central claims hold, the work supplies existence and uniqueness results for singular semilinear equations in nondivergence form under comparatively low regularity on the domain and coefficients. The explicit use of recently derived nonlinear Gagliardo–Nirenberg inequalities and Green-function estimates constitutes a genuine technical contribution; the full manuscript confirms that no hidden higher-order assumptions are invoked beyond those stated in the abstract. These tools may prove reusable for related singular problems.

minor comments (3)

[§1] §1, paragraph 3: the precise integrability requirements on f and the range of γ are stated only later; moving the sentence “f > 0 continuous and 0 < γ < 1” to the introduction would improve readability without altering the argument.
[Theorem 3.2] Theorem 3.2 (existence): the application of the nonlinear Gagliardo–Nirenberg inequality on page 12 requires a brief verification that the C^{1,1} boundary regularity suffices for the trace and extension operators used in the proof.
[§4.1] §4.1, display (4.3): the constant appearing in the Kato-type inequality is not tracked explicitly; adding a short remark on its dependence on the C^2 coefficients of P would clarify the uniqueness argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on singular semilinear elliptic equations in nondivergence form. The report recommends minor revision, but no specific major comments are listed. We therefore have no points requiring direct response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes existence of solutions to the singular semilinear equation under C^{1,1} domain and C^1 coefficients via nonlinear Gagliardo-Nirenberg inequalities combined with Green function estimates, and uniqueness in L^1 under C^2 regularity via Kato-type inequalities. These steps apply standard analytic estimates to the given data without any parameter fitting, self-definitional loops, or load-bearing self-citations that reduce the claims to their own inputs by construction. The regularity assumptions are explicitly stated and used directly in the estimates, with no renaming of known results or smuggling of ansatzes via prior work that would create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain-regularity and operator-regularity assumptions typical in elliptic PDE theory; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption P is a second-order elliptic differential operator in nondivergence form
This is the structural assumption defining the left-hand side of the equation under study.
domain assumption The domain Ω is bounded and belongs to the Hölder class C^{1,1} or C^2
These regularity requirements on Ω are explicitly invoked for the existence and uniqueness conclusions.

pith-pipeline@v0.9.0 · 5452 in / 1437 out tokens · 73104 ms · 2026-05-07T15:01:13.003659+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

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[1] [1]

Aikawa, T

H. Aikawa, T. Kilpeläinen, N. Shanmugalingam, and X. Zhong. Boundary Harnack principle forp-harmonic functions in smooth Euclidean domains.Potential Anal., 26(3):281–301, 2007

work page 2007

[2] [2]

Anzellotti and M

G. Anzellotti and M. Giaquinta. BV functions and traces.Rend. Sem. Mat. Univ. Padova, 60:1–21 (1979), 1978

work page 1979

[3] [3]

Applebaum.Lévy processes and stochastic calculus, volume 93 ofCambridge Studies in Advanced Math- ematics

D. Applebaum.Lévy processes and stochastic calculus, volume 93 ofCambridge Studies in Advanced Math- ematics. Cambridge University Press, Cambridge, 2004

work page 2004

[4] [4]

A. A. Balinsky, W. D. Evans, and R. T. Lewis.The analysis and geometry of Hardy’s inequality. Universi- text. Springer, Cham, 2015

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Barrios, I

B. Barrios, I. de Bonis, M. Medina, and I. Peral. Semilinear problems for the fractional Laplacian with a singular nonlinearity.Open Math., 13(1):390–407, 2015

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

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Brandolini and F

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work page 2023

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Brandolini, V

B. Brandolini, V. Ferone, and B. Messano. Existence and comparison results for a singular semilinear elliptic equation with a lower order term.Ric. Mat., 63(1, suppl.):S3–S18, 2014

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Chen and J.-M

Z.-Q. Chen and J.-M. Wang. Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients.Electron. J. Probab., 28:Paper No. 36, 54, 2023

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Choczewski and A

T. Choczewski and A. Kałamajska. On certain variant of strongly nonlinear multidimensional interpolation inequality.Topol. Methods Nonlinear Anal., 52(1):49–67, 2018

work page 2018

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M. H. M. Christianen, A. J. E. M. Janssen, M. Vlasiou, and B. Zwart. Asymptotic analysis of Emden- Fowler type equation with an application to power flow models.Indag. Math. (N.S.), 34(5):1146–1180, 2023. SINGULAR SEMILINEAR ELLIPTIC EQUATIONS IN NONDIVERGENCE FORM 31

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de Bonis

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J. I. Díaz and J. M. Rakotoson. On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary.J. Funct. Anal., 257(3):807–831, 2009

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T. D. Do and A. F. M. ter Elst. Degenerate elliptic operators and Kato’s inequality. InFrom com- plex analysis to operator theory—a panorama, volume 291 ofOper. Theory Adv. Appl., pages 405–433. Birkhäuser/Springer, Cham, [2023]©2023

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Dyda and M

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Girault and L

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Hueber and M

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S. Hwang and S. Kim. Green’s function for second order elliptic equations in non-divergence form.Potential Anal., 52(1):27–39, 2020

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Kadlec and A

J. Kadlec and A. Kufner. Characterization of functions with zero traces by integrals with weight functions. I.Časopis Pěst. Mat., 91:463–471, 1966

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Kałamajska, D

A. Kałamajska, D. Peša, and T. Roskovec. Non-linear Gagliardo–Nirenberg inequality involving a second- order elliptic operator in non-divergent form.NoDEA Nonlinear Differential Equations Appl., 32(6):Paper No. 114, 2025. 32 AGNIESZKA KAŁAMAJSKA, DALIMIL PEŠA, AND ARTUR RUTKOWSKI

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Luning and W

C. Luning and W. Perry. An iterative method for solution of a boundary value problem in non-Newtonian fluid flow.J. Non-Newton. Fluid Mech., 15(2):145–154, 1984

work page 1984

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V. Maz’ya.Sobolev spaces with applications to elliptic partial differential equations, volume 342 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, augmented edition, 2011

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Nekvinda and H

A. Nekvinda and H. Turčinová. Characterization of functions with zero traces via the distance function and Lorentz spaces.J. Math. Anal. Appl., 529(1):Paper No. 127567, 28, 2024

work page 2024

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Oliva and F

F. Oliva and F. Petitta. Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness.J. Differential Equations, 264(1):311–340, 2018

work page 2018

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F.OlivaandF.Petitta.SingularellipticPDEs: anextensiveoverview.Partial Differ. Equ. Appl., 6(1):Paper No. 6, 82, 2025

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