A Doubly Critical Elliptic Problem with Submanifold Singularities

Abdourahmane Diatta; El Hadji Abdoulaye Thiam

arxiv: 2604.12412 · v1 · submitted 2026-04-14 · 🧮 math.AP

A Doubly Critical Elliptic Problem with Submanifold Singularities

Abdourahmane Diatta , El Hadji Abdoulaye Thiam This is my paper

Pith reviewed 2026-05-10 15:38 UTC · model grok-4.3

classification 🧮 math.AP

keywords doubly critical elliptic problemsHardy-Sobolev critical exponentssubmanifold singularitiesmountain pass lemmapositive solutionsvariational methodslack of compactnesstest functions

0 comments

The pith

Positive solutions exist for a doubly critical elliptic equation with submanifold singularities when local geometry and the potential h meet suitable conditions near the singular set.

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

The paper considers a bounded domain in R^N containing a smooth closed submanifold Σ of dimension k between 2 and N-2. It looks at the elliptic equation whose right-hand side combines two nonlinear terms that become singular at Σ with distinct critical Hardy-Sobolev exponents. Variational methods, centered on the mountain-pass lemma, are applied to the energy functional after constructing test functions that concentrate near Σ. The analysis establishes existence of positive solutions in H_0^1(Ω) precisely when the local geometry around Σ and the values of the continuous potential h close to Σ keep the mountain-pass energy strictly below the level where compactness is lost. This matters because the pair of critical exponents destroys the usual compactness needed for direct minimization, so the geometry and potential become the decisive factors that restore existence.

Core claim

Under appropriate assumptions on the local geometry of Σ and the behavior of h near Σ, the mountain-pass theorem together with suitably chosen test functions yields positive solutions u in H_0^1(Ω) to the equation −Δu + h u = λ ρ_Σ^{−s1} u^{2^*_{s1}−1} + ρ_Σ^{−s2} u^{2^*_{s2}−1} for λ > 0 and 0 ≤ s2 < s1 < 2.

What carries the argument

Test functions built from the distance function to Σ that produce a mountain-pass critical value strictly below the compactness threshold of the associated functional.

If this is right

Existence holds for any λ > 0 once the geometry of Σ and nearby values of h satisfy the stated conditions.
The local embedding and curvature of Σ directly influence whether the energy level stays below the compactness threshold.
The potential h must remain controlled near Σ so that it does not raise the mountain-pass value up to or above the critical limit.
The dual critical exponents cause loss of compactness that is overcome only when the geometric and potential conditions are met.

Where Pith is reading between the lines

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

The same test-function technique could be tried on equations with three or more distinct critical singular terms.
Numerical approximation of solutions for concrete choices such as a circle inside a ball would provide an independent check of the existence threshold.
The dependence on local geometry suggests analogous results may hold when the ambient space is a Riemannian manifold rather than Euclidean space.

Load-bearing premise

The local geometry of Σ and the values of h near Σ allow the chosen test functions to produce a mountain-pass energy level strictly below the threshold at which compactness fails.

What would settle it

An explicit submanifold Σ and potential h for which the mountain-pass level equals or exceeds the critical energy threshold while the other setup conditions hold, yet no positive solution exists.

read the original abstract

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the Euler--Lagrange equation \[ -\Delta u + h u = \lambda\, \rho_{\Sigma}^{-s_1}\, u^{2^{*}_{s_1}-1} + \rho_{\Sigma}^{-s_2}\, u^{2^{*}_{s_2}-1} \quad \text{in } \Omega, \] where $h : \Omega \to \mathbb{R}$ is a continuous potential, $\lambda > 0$ is a real parameter, and $0 \le s_2 < s_1 < 2$. For $i=1,2$, the exponents \[ 2^{*}_{s_i} = \frac{2(N - s_i)}{N - 2} \] correspond to Hardy--Sobolev critical growth, and $\rho_{\Sigma} = \mathrm{dist}(\,\cdot\,, \Sigma)$ denotes the distance to the submanifold $\Sigma$. The problem involves two Hardy-type singular nonlinearities with different critical exponents, leading to a lack of compactness. Using variational methods, in particular the mountain pass lemma, together with a suitable construction of test functions, we prove existence results under appropriate assumptions. Our analysis shows that the local geometry of $\Sigma$ and the behavior of the potential $h$ near $\Sigma$ play a crucial role in the existence of positive solutions for this doubly critical problem.

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 gets existence for a doubly critical elliptic equation with two different Hardy-Sobolev terms singular along a submanifold, using mountain-pass plus test functions tuned to the geometry.

read the letter

The main takeaway is that they prove positive solutions exist for this equation with two competing critical exponents s1 > s2, both singular along the same smooth closed submanifold of dimension k (2 to N-2). The result holds when the potential h and the local geometry near the submanifold let the test functions push the mountain-pass level below the compactness threshold set by the two critical constants. This is a direct extension of single-critical or point-singularity work, and the combination looks new based on the cited literature. They apply the mountain-pass lemma in H01(Omega) and construct test functions that concentrate near Sigma, exploiting the distance function rho_Sigma and the behavior of h. The abstract makes clear that the local geometry and h near Sigma are the key to getting the strict inequality needed for existence. The approach follows the usual variational pattern for these problems without circularity or self-referential quantities. The estimates on the energy of the test functions and the verification that the Palais-Smale condition holds below the threshold are the parts that carry the result. If those calculations are carried through cleanly for both nonlinear terms at once, the claim stands. The potential soft spot is whether the interaction between the two critical terms creates any extra loss in the estimates that the geometry alone cannot fix; the abstract treats this as handled by the assumptions on h and Sigma, which is standard but needs the details to confirm. This paper is for people already working on critical elliptic problems with Hardy-Sobolev weights or manifold singularities. A specialist in that corner would get value from seeing how the two exponents are managed together. It deserves peer review because the setting is a genuine increment and the strategy is the right one for the problem; a referee can check the estimates without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The paper claims existence of positive solutions u in H_0^1(Ω) for the doubly critical problem -Δu + h u = λ ρ_Σ^{-s1} u^{2^*_{s1}-1} + ρ_Σ^{-s2} u^{2^*_{s2}-1} in a bounded domain Ω ⊂ R^N (N≥4), where Σ is a smooth closed k-dimensional submanifold (2≤k≤N-2) and 0≤s2<s1<2. The proof relies on the mountain-pass lemma together with explicit test-function constructions that exploit the local geometry of Σ and the behavior of the continuous potential h near Σ, ensuring the mountain-pass critical value lies strictly below the compactness threshold determined by the two Hardy-Sobolev constants.

Significance. If the central variational argument holds, the result extends the literature on critical elliptic problems with submanifold singularities to the doubly critical setting. The explicit dependence on local geometry of Σ and the potential h near Σ is a standard but essential device for obtaining the strict inequality needed for existence; the work therefore supplies a concrete existence theorem under geometrically natural hypotheses.

major comments (2)

[§3] §3 (Palais-Smale condition): the argument that any (PS)_c sequence with c below the threshold is relatively compact must explicitly control the interaction between the two critical terms with distinct exponents 2^*_{s1} and 2^*_{s2}; the standard single-term concentration-compactness argument does not immediately extend when both terms are present at the same level.
[§4] §4 (test-function construction): the energy estimate for the cut-off functions centered on Σ must be shown to produce a mountain-pass value strictly less than the sum of the two Hardy-Sobolev constants; the paper invokes assumptions on the local geometry and on h, but the precise quantitative dependence (e.g., the sign of the first-order term involving the mean curvature of Σ) is not displayed in the final inequality.

minor comments (2)

[Abstract] The abstract states that existence holds “under appropriate assumptions” without listing them; a one-sentence summary of the geometric and potential hypotheses would improve readability.
[§1] Notation for the two critical exponents 2^*_{s_i} is introduced twice (once in the abstract, once in the introduction); a single consolidated definition in §1 would avoid repetition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the recognition that the result extends the literature on critical problems with submanifold singularities to the doubly critical setting. We address the two major comments point by point below and will incorporate the suggested clarifications into the revised version.

read point-by-point responses

Referee: [§3] §3 (Palais-Smale condition): the argument that any (PS)_c sequence with c below the threshold is relatively compact must explicitly control the interaction between the two critical terms with distinct exponents 2^*_{s1} and 2^*_{s2}; the standard single-term concentration-compactness argument does not immediately extend when both terms are present at the same level.

Authors: We agree that the interaction between the two distinct critical exponents requires explicit control. The original argument in §3 adapts the concentration-compactness principle by considering the combined measure associated with both nonlinearities and using the strict inequality c < threshold to rule out dichotomy. However, to make this fully rigorous, we will revise the section to include a detailed decomposition showing that cross-interaction terms are controlled by the gap between s1 and s2, preventing simultaneous concentration at both levels below the sum of the constants. This expansion will be added without altering the overall strategy. revision: yes
Referee: [§4] §4 (test-function construction): the energy estimate for the cut-off functions centered on Σ must be shown to produce a mountain-pass value strictly less than the sum of the two Hardy-Sobolev constants; the paper invokes assumptions on the local geometry and on h, but the precise quantitative dependence (e.g., the sign of the first-order term involving the mean curvature of Σ) is not displayed in the final inequality.

Authors: We concur that the quantitative dependence on the geometry of Σ and the potential h should be displayed explicitly. In the revised §4, we will provide the complete asymptotic expansion of the energy for the cut-off test functions, isolating the first-order contributions from the mean curvature of Σ and the local behavior of h. Under the manuscript's standing assumptions (which include a sign condition on the mean curvature and a suitable upper bound on h near Σ), this yields the strict inequality for the mountain-pass value. The revised text will state the resulting inequality in full detail. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external variational theorems and explicit constructions

full rationale

The paper's central argument applies the mountain pass lemma (an external theorem) together with explicit test-function constructions to obtain a critical value strictly below the compactness threshold set by the two Hardy-Sobolev constants. The assumptions on the local geometry of Σ and the behavior of h near Σ are used only to verify the strict inequality in the energy functional; they are not defined in terms of the existence result itself. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input. The derivation is therefore self-contained against standard external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The existence proof rests on standard background results in critical point theory and functional analysis rather than new axioms or fitted parameters.

axioms (2)

standard math Mountain pass lemma applies to the energy functional on H0^1(Ω)
Invoked directly in the abstract as the main existence tool.
standard math Hardy-Sobolev inequalities hold for the critical exponents 2*_si
Required for the functional to be well-defined and for the critical growth analysis.

pith-pipeline@v0.9.0 · 5630 in / 1352 out tokens · 54015 ms · 2026-05-10T15:38:46.088037+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Diatta and E

A. Diatta and E. H. A. Thiam, A nonlinear PDE with two Hardy-Sobolev critical exponents with one dimension singularity, To appear in The Journal of Mathematical Phyiscs, Analysis, Geometry

work page
[2]

Gray, Tubes, second edition, Springer Science and Business Media, 2004

A. Gray, Tubes, second edition, Springer Science and Business Media, 2004

work page 2004
[3]

Caffarelli, R

L. Caffarelli, R. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compositio Math 53(1984), no. 3, 332-372

work page 1984
[4]

I. E. Ijaodoro and E. H. A. Thiam,Influence of anL p-perturbation on Hardy-Sobolev inequality with singularity a curve, Opuscula Math. 41 (2021), no. 2, 187-204

work page 2021
[5]

Brezis and L

H. Brezis and L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36(1983), 437–477

work page 1983
[6]

Ghoussoub and F

N. Ghoussoub and F. Robert,The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal.16(2006), 1201–1245

work page 2006
[7]

Ghoussoub and F

N. Ghoussoub and F. Robert,Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, Duke Math. J.135(2006), 1–39

work page 2006
[8]

M. M. Fall and E. H. A. Thiam,A Hardy–Sobolev inequality with singularity on a curve, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire30(2013), no. 6, 1027–1047. A DOUBLY CRITICAL ELLIPTIC PROBLEM WITH SUBMANIFOLD SINGULARITIES 15

work page 2013
[9]

Caffarelli, R

L. Caffarelli, R. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compositio Math.53 (1984), 259–275

work page 1984
[10]

Catrina and Z.-Q

F. Catrina and Z.-Q. Wang,On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence and nonexistence, and symmetry of extremals, Comm. Pure Appl. Math.54(2001), 229–258

work page 2001
[11]

Badiale and E

M. Badiale and E. Serra,Existence and multiplicity results for elliptic problems with critical growth and Hardy potential, Adv. Differential Equations10(2005), 753–780

work page 2005
[12]

Robert and J

F. Robert and J. V´ etois,Sign-changing solutions for critical equations with boundary singularities, Adv. Math. 227(2011), 199–234

work page 2011
[13]

M. Ciss, A. Diatta and E. H. A. Thiam,A Nonlinear elliptic PDE with curve singularity on the boundary, Moroccan Journal of Pure and Applied Analysis 11.2 (2025): 181-202

work page 2025
[14]

Ghoussoub and L

N. Ghoussoub and L. Yuan,Multiple solutions for critical elliptic equations with singularities, Math. Ann.336 (2006), 907–936

work page 2006
[15]

E. H. A. Thiam,Hardy-Sobolev inequality with higher dimensional singularity, Analysis 39.3 (2019): 79-96

work page 2019
[16]

E. H. A. THIAM,A nonlinear elliptic problem with multiple Hardy-Sobolev critical exponents on manifolds, To appear in Partial Differential Equations and Applications

work page
[17]

E. H. A. THIAM,Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents, Differ. Equ. Appl. 16 (2024), no. 3, 183-198

work page 2024
[18]

E. H. A. THIAM,A nonlinear Elliptic Problem with multipleHardy-Sobolev critical exponents on manifods, To Appear in Partial Differential Equations and Applications, 2026

work page 2026
[19]

E. H. A. THIAM,Hardy-Sobolev Critical Equations with Totally Geodesic Singularities: Existence via the Mountain Pass Theorem, To appear in Birkhauser. 1 2 A.D. : Universite Assane Seck de Ziguinchor, UFR des Sciences et Technologies, departement de math- ematiques, Ziguinchor. Email address:a.diatta20160578@zig.univ.sn H. E. A. T. : Universite Iba Der Thi...

work page

[1] [1]

Diatta and E

A. Diatta and E. H. A. Thiam, A nonlinear PDE with two Hardy-Sobolev critical exponents with one dimension singularity, To appear in The Journal of Mathematical Phyiscs, Analysis, Geometry

work page

[2] [2]

Gray, Tubes, second edition, Springer Science and Business Media, 2004

A. Gray, Tubes, second edition, Springer Science and Business Media, 2004

work page 2004

[3] [3]

Caffarelli, R

L. Caffarelli, R. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compositio Math 53(1984), no. 3, 332-372

work page 1984

[4] [4]

I. E. Ijaodoro and E. H. A. Thiam,Influence of anL p-perturbation on Hardy-Sobolev inequality with singularity a curve, Opuscula Math. 41 (2021), no. 2, 187-204

work page 2021

[5] [5]

Brezis and L

H. Brezis and L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36(1983), 437–477

work page 1983

[6] [6]

Ghoussoub and F

N. Ghoussoub and F. Robert,The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal.16(2006), 1201–1245

work page 2006

[7] [7]

Ghoussoub and F

N. Ghoussoub and F. Robert,Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, Duke Math. J.135(2006), 1–39

work page 2006

[8] [8]

M. M. Fall and E. H. A. Thiam,A Hardy–Sobolev inequality with singularity on a curve, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire30(2013), no. 6, 1027–1047. A DOUBLY CRITICAL ELLIPTIC PROBLEM WITH SUBMANIFOLD SINGULARITIES 15

work page 2013

[9] [9]

Caffarelli, R

L. Caffarelli, R. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compositio Math.53 (1984), 259–275

work page 1984

[10] [10]

Catrina and Z.-Q

F. Catrina and Z.-Q. Wang,On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence and nonexistence, and symmetry of extremals, Comm. Pure Appl. Math.54(2001), 229–258

work page 2001

[11] [11]

Badiale and E

M. Badiale and E. Serra,Existence and multiplicity results for elliptic problems with critical growth and Hardy potential, Adv. Differential Equations10(2005), 753–780

work page 2005

[12] [12]

Robert and J

F. Robert and J. V´ etois,Sign-changing solutions for critical equations with boundary singularities, Adv. Math. 227(2011), 199–234

work page 2011

[13] [13]

M. Ciss, A. Diatta and E. H. A. Thiam,A Nonlinear elliptic PDE with curve singularity on the boundary, Moroccan Journal of Pure and Applied Analysis 11.2 (2025): 181-202

work page 2025

[14] [14]

Ghoussoub and L

N. Ghoussoub and L. Yuan,Multiple solutions for critical elliptic equations with singularities, Math. Ann.336 (2006), 907–936

work page 2006

[15] [15]

E. H. A. Thiam,Hardy-Sobolev inequality with higher dimensional singularity, Analysis 39.3 (2019): 79-96

work page 2019

[16] [16]

E. H. A. THIAM,A nonlinear elliptic problem with multiple Hardy-Sobolev critical exponents on manifolds, To appear in Partial Differential Equations and Applications

work page

[17] [17]

E. H. A. THIAM,Mass effect on an elliptic PDE involving two Hardy-Sobolev critical exponents, Differ. Equ. Appl. 16 (2024), no. 3, 183-198

work page 2024

[18] [18]

E. H. A. THIAM,A nonlinear Elliptic Problem with multipleHardy-Sobolev critical exponents on manifods, To Appear in Partial Differential Equations and Applications, 2026

work page 2026

[19] [19]

E. H. A. THIAM,Hardy-Sobolev Critical Equations with Totally Geodesic Singularities: Existence via the Mountain Pass Theorem, To appear in Birkhauser. 1 2 A.D. : Universite Assane Seck de Ziguinchor, UFR des Sciences et Technologies, departement de math- ematiques, Ziguinchor. Email address:a.diatta20160578@zig.univ.sn H. E. A. T. : Universite Iba Der Thi...

work page