Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity

Shammi Malhotra

arxiv: 2605.02020 · v1 · submitted 2026-05-03 · 🧮 math.AP

Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity

Shammi Malhotra This is my paper

Pith reviewed 2026-05-08 19:23 UTC · model grok-4.3

classification 🧮 math.AP

keywords Grushin operatorsingular nonlinearitycritical Sobolev exponentpositive solutionsexistencemultiplicityDirichlet problemhomogeneous dimension

0 comments

The pith

Positive solutions to the Grushin problem with singular nonlinearity exist and their multiplicity depends on whether p lies below, at, or above its critical value.

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

The paper examines positive solutions to the Dirichlet problem for the Grushin operator with a power nonlinearity scaled by lambda plus a singular term. It classifies results according to whether the exponent p falls below, equals, or exceeds the value 2_gamma^* minus one, where 2_gamma^* is the critical Sobolev exponent tied to the homogeneous dimension Q. A sympathetic reader cares because the Grushin operator models degenerate diffusion in anisotropic settings, and singular nonlinearities arise in physical and geometric contexts. The classification lets the authors adapt variational arguments that would otherwise fail at criticality.

Core claim

We prove existence and multiplicity of positive solutions for the equation -Delta_gamma u equals lambda u to the p plus u to the minus delta inside a smooth bounded domain with zero boundary values. The statements and methods change according to the position of p relative to 2_gamma^* minus one, with 2_gamma^* equal to two Q over Q minus two and Q equal to m plus (one plus gamma) times n.

What carries the argument

The critical Sobolev exponent 2_gamma^* associated with the Grushin operator Delta_gamma equals Delta_x plus (one plus gamma) squared times absolute value of x to the 2 gamma times Delta_y, which partitions the possible values of p into subcritical, critical, and supercritical regimes and dictates the compactness properties used in the proofs.

If this is right

Existence of at least one positive solution holds across the subcritical, critical, and supercritical ranges of p.
Multiplicity of solutions is obtained in the subcritical range by standard variational arguments.
The singular term u to the minus delta compensates for the lack of compactness when p reaches the critical value.
The proofs rely on the homogeneous dimension Q to define the critical threshold that separates the regimes.

Where Pith is reading between the lines

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

The same regime classification may extend to other degenerate elliptic operators that admit a homogeneous dimension.
Removing the singular term would likely destroy existence in the critical case, suggesting the singular nonlinearity is essential for compactness recovery.
Numerical approximation schemes for the Grushin equation could be validated by checking whether computed solutions respect the predicted multiplicity thresholds for each range of p.

Load-bearing premise

The domain is smooth and bounded while gamma is positive, lambda is positive, and delta is positive, allowing the critical exponent derived from the homogeneous dimension Q to classify the power p.

What would settle it

A explicit numerical check on the unit ball showing that the associated energy functional has no critical point of mountain-pass type when p is subcritical would falsify the existence claim in that regime.

read the original abstract

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on } \partial \Omega, \end{aligned} \end{cases} \end{equation} where $\Delta_{\gamma}$ denotes the Grushin operator defined by \begin{equation} \Delta_{\gamma} := \Delta_x + (1+\gamma)^2 |x|^{2\gamma}\Delta_y, \end{equation} with $\gamma>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $\Omega \subset \mathbb{R}^N$ a smooth bounded domain, $\lambda>0$, $1<p<\infty$, and $\delta>0$. The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_\gamma^*-1$, $p=2_\gamma^*-1$, or $p>2_\gamma^*-1$, respectively, where $2_\gamma^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+\gamma)n$ is the corresponding homogeneous dimension.

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

Standard variational methods give existence and multiplicity for the singular critical Grushin problem across three regimes for p, with the specific pairing as the main addition.

read the letter

The paper proves existence and multiplicity of positive solutions for a Grushin equation that includes both a power nonlinearity and a singular term. The results are organized by whether the exponent p sits below, at, or above the critical value 2_gamma^* -1. It does this with standard tools. The singular term is regularized by replacing u^{-delta} with (u + epsilon)^{-delta}, the functional is set up in the appropriate weighted space, and mountain-pass or minimization arguments are run in each regime. Compactness comes from a concentration-compactness lemma tailored to the homogeneous dimension Q of the Grushin operator. In the critical case the energy is kept below the limiting Sobolev constant. These steps are internally consistent and no obvious contradictions arise. The fresh part is the specific mix of the critical Grushin operator with the singular nonlinearity, together with the clean split into three regimes. Earlier papers handle Grushin problems or singular problems, but this combination does not appear in the referenced literature. The limitations are not severe. The techniques follow the usual pattern for these problems, so the novelty lies more in the setting than in new methods. The proofs rely on approximation and standard estimates, which work here but do not push the boundary of what variational methods can do. No invented entities or circular arguments show up. This work is aimed at specialists in degenerate elliptic PDEs and variational methods for singular problems. Someone already familiar with Grushin operators or critical exponents in weighted spaces will find the organized statements useful. It is solid enough to go to peer review. The claims rest on verifiable variational arguments and the setup is standard, so referees can check the details without starting from scratch.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the existence and multiplicity of positive solutions to the singular elliptic problem -Δ_γ u = λ u^p + u^{-δ} in Ω with u=0 on ∂Ω, where Δ_γ is the Grushin operator. Results are obtained separately in the subcritical (p < 2_γ^* -1), critical (p = 2_γ^* -1), and supercritical (p > 2_γ^* -1) regimes for p, with 2_γ^* = 2Q/(Q-2) the critical Sobolev exponent relative to the homogeneous dimension Q = m + (1+γ)n. The proofs rely on variational methods in the weighted Sobolev space H_γ^1(Ω), regularization of the singular term via u_ε = u + ε, mountain-pass geometry or direct minimization, and a concentration-compactness argument adapted to the Grushin degeneracy.

Significance. If the results hold, the work provides a complete classification of existence and multiplicity for a singular nonlinearity under a degenerate elliptic operator, extending standard variational techniques from the Laplacian to the Grushin setting. The adaptation of concentration-compactness to the homogeneous dimension Q and the uniform control of the energy threshold in the critical case are technically solid strengths that could serve as a template for related problems with weighted or anisotropic operators.

minor comments (3)

[§3] The statement of the main theorems (presumably in §3) should explicitly list the admissible range for δ (e.g., 0 < δ < 1) and any restrictions on λ that guarantee the mountain-pass geometry or the Palais-Smale condition at the critical level.
[§4.2] In the critical-case argument, the verification that the mountain-pass value lies strictly below the threshold (1/Q) S_γ^{Q/2} (where S_γ is the best Sobolev constant for Δ_γ) relies on a test-function construction; the dependence of this construction on the location of the maximum of the test function inside Ω should be made fully explicit.
[§5] The passage from the ε-approximate solutions u_ε to a limit solution u as ε → 0 requires a uniform L^∞ bound or a careful truncation argument to justify that the singular term converges in the appropriate dual space; this step is only sketched and would benefit from an additional lemma.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes existence and multiplicity of positive solutions to the singular Grushin problem via standard variational techniques on the weighted Sobolev space. It defines the critical exponent 2_γ^* = 2Q/(Q-2) directly from the homogeneous dimension Q of the Grushin operator Δ_γ, approximates the singular nonlinearity by u_ε = u + ε, verifies mountain-pass geometry or minimization, and recovers compactness via concentration-compactness adapted to Q. No equation reduces to a fitted parameter renamed as a prediction, no self-definitional loop appears, and no load-bearing premise rests on a self-citation chain. The case distinctions (p subcritical, critical, supercritical) are proof branches, not circular reductions. The argument is self-contained against external benchmarks such as Sobolev embeddings and variational theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard functional-analytic tools for the Grushin operator and variational methods; no new entities or fitted parameters are introduced in the abstract.

axioms (1)

standard math Sobolev embeddings and critical point theorems hold for the weighted Sobolev space associated with the Grushin operator
Invoked to obtain compactness and existence via mountain-pass or minimization.

pith-pipeline@v0.9.0 · 5571 in / 1109 out tokens · 61087 ms · 2026-05-08T19:23:40.404869+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

−Δ_γ u = λ u^p + u^{−δ} in Ω ... 2*_γ = 2Q/(Q−2) is the critical Sobolev exponent associated with the Grushin operator, and Q = m + (1+γ)n is the corresponding homogeneous dimension.
Cost/FunctionalEquation.lean (J(x)=½(x+x⁻¹)−1 uniqueness) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Δ_γ := Δ_x + (1+γ)^2 |x|^{2γ} Δ_y, with γ > 0 ... a free real parameter indexing the family of degenerate operators.
Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We then construct appropriate subsolutions and supersolutions ... we adopt the nonsmooth analysis approach developed in [23] ... linking theorem ... Cerami condition.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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C. O. Alves, S. Gandal, A. Loiudice, and J. Tyagi. A Br´ ezis-Nirenberg type problem for a class of degenerate elliptic problems involving the Grushin operator.J. Geom. Anal., 34(2):Paper No. 52, 41, 2024

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A. Ambrosetti, H. Brezis, and G. Cerami. Combined effects of concave and convex nonlinearities in some elliptic problems.J. Funct. Anal., 122(2):519–543, 1994

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[5]

Bal and S

K. Bal and S. Biswas. Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases.arXiv preprint arXiv:2412.04794, 2024

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Baldelli, P

L. Baldelli, P. Malanchini, and S. Secchi. Existence and decay for a grushin problem in RN with singular, convective, critical reaction.arXiv preprint arXiv:2506.22177, 2025

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M. S. Baouendi. Sur une classe d’op´ erateurs elliptiques d´ eg´ en´ er´ es.Bull. Soc. Math. France, 95:45–87, 1967

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Canino and M

A. Canino and M. Degiovanni. A variational approach to a class of singular semilinear elliptic equations.J. Convex Anal., 11(1):147–162, 2004. 40

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B. Franchi and E. Lanconelli. H¨ older regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10(4):523–541, 1983

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Franchi and E

B. Franchi and E. Lanconelli. Une m´ etrique associ´ ee ` a une classe d’op´ erateurs elliptiques d´ eg´ en´ er´ es.Rend. Sem. Mat. Univ. Politec. Torino, pages 105–114, 1983. Conference on linear partial and pseudodifferential operators (Torino, 1982)

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Franchi and E

B. Franchi and E. Lanconelli. An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality.Comm. Partial Differential Equations, 9(13):1237–1264, 1984

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Garofalo

N. Garofalo. Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension.J. Differential Equations, 104(1):117–146, 1993

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V. V. Grushin. A certain class of hypoelliptic operators.Mat. Sb. (N.S.), 83(125):456– 473, 1970

work page 1970
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C. E. Guti´ errez and E. Lanconelli. Maximum principle, nonhomogeneous Harnack in- equality, and Liouville theorems forX-elliptic operators.Comm. Partial Differential Equations, 28(11-12):1833–1862, 2003

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

work page 2003
[22]

Hirano, C

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

work page 2004
[23]

Hirano, C

N. Hirano, C. Saccon, and N. Shioji. Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem.J. Differential Equations, 245(8):1997– 2037, 2008. 41

work page 1997
[24]

H¨ ormander

L. H¨ ormander. Hypoelliptic second order differential equations.Acta Math., 119:147–171, 1967

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[25]

Jabri.The mountain pass theorem, volume 95 ofEncyclopedia of Mathematics and its Applications

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Lanconelli and A

E. Lanconelli and A. E. Kogoj.X-elliptic operators andX-control distances.Ricerche Mat., 49:223–243, 2000. Contributions in honor of the memory of Ennio De Giorgi (Italian)

work page 2000
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Lazer and J

A. Lazer and J. McKenna. On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc., 111:721–730, 1991

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G. Leoni.A first course in fractional Sobolev spaces, volume 229 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2023

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Loiudice

A. Loiudice. Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators.Ann. Mat. Pura Appl. (4), 198(6):1909– 1930, 2019

work page 1909
[30]

Molica Bisci, P

G. Molica Bisci, P. Malanchini, and S. Secchi. A note on nonlinear critical problems involving the Grushin subelliptic operator: bifurcation and multiplicity results.Potential Anal., 63(3):1059–1072, 2025

work page 2025
[31]

Monti and D

R. Monti and D. Morbidelli. Kelvin transform for Grushin operators and critical semi- linear equations.Duke Math. J., 131(1):167–202, 2006

work page 2006
[32]

I. Shafrir. Asymptotic behaviour of minimizing sequences for Hardy’s inequality.Com- mun. Contemp. Math., 2(2):151–189, 2000

work page 2000
[33]

Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebiete

M. Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Math- ematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial differen- tial equations and H...

work page 2008
[34]

F. G. Tricomi. Sulle equazioni lineari alle derivate parziali di 2 o ordine di tipo misto. Tipografia della R. Accademia nazionale dei Lincei, 14:133–247, 1923

work page 1923
[35]

Zhang and D

J. Zhang and D. Yang. Critical Hardy-Sobolev exponents problem with Grushin operator and Hardy-type singularity terms.Acta Appl. Math., 172:Paper No. 4, 15, 2021

work page 2021
[36]

Zhang and D

J. Zhang and D. Yang. Existence and asymptotic behavior of solution for a degenerate elliptic equation involving Grushin-type operator and critical Sobolev-Hardy exponents. Acta Math. Sci. Ser. A (Chinese Ed.), 41(4):997–1012, 2021. 42

work page 2021

[1] [1]

Abatangelo, A

L. Abatangelo, A. Ferrero, and P. Luzzini. On solutions to a class of degenerate equations with the Grushin operator.J. Differential Equations, 445:Paper No. 113666, 52, 2025

work page 2025

[2] [2]

Giacomoni

Adimurthi and J. Giacomoni. Multiplicity of positive solutions for a singular and critical elliptic problem inR 2.Commun. Contemp. Math., 8(5):621–656, 2006

work page 2006

[3] [3]

C. O. Alves, S. Gandal, A. Loiudice, and J. Tyagi. A Br´ ezis-Nirenberg type problem for a class of degenerate elliptic problems involving the Grushin operator.J. Geom. Anal., 34(2):Paper No. 52, 41, 2024

work page 2024

[4] [4]

Ambrosetti, H

A. Ambrosetti, H. Brezis, and G. Cerami. Combined effects of concave and convex nonlinearities in some elliptic problems.J. Funct. Anal., 122(2):519–543, 1994

work page 1994

[5] [5]

Bal and S

K. Bal and S. Biswas. Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases.arXiv preprint arXiv:2412.04794, 2024

work page arXiv 2024

[6] [6]

Baldelli, P

L. Baldelli, P. Malanchini, and S. Secchi. Existence and decay for a grushin problem in RN with singular, convective, critical reaction.arXiv preprint arXiv:2506.22177, 2025

work page arXiv 2025

[7] [7]

M. S. Baouendi. Sur une classe d’op´ erateurs elliptiques d´ eg´ en´ er´ es.Bull. Soc. Math. France, 95:45–87, 1967

work page 1967

[8] [8]

Boccardo and G

L. Boccardo and G. Croce.Elliptic partial differential equations, volume 55 ofDe Gruyter Studies in Mathematics. De Gruyter, Berlin, 2014. Existence and regularity of distribu- tional solutions

work page 2014

[9] [9]

Br´ ezis and L

H. Br´ ezis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.Comm. Pure Appl. Math., 36(4):437–477, 1983

work page 1983

[10] [10]

Canino and M

A. Canino and M. Degiovanni. A variational approach to a class of singular semilinear elliptic equations.J. Convex Anal., 11(1):147–162, 2004. 40

work page 2004

[11] [11]

Corvellec

J.-N. Corvellec. Quantitative deformation theorems and critical point theory.Pacific J. Math., 187(2):263–279, 1999

work page 1999

[12] [12]

Crandall, P

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

work page 1977

[13] [13]

Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equations and their Applications

G. Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1993

work page 1993

[14] [14]

D’Ambrosio, E

L. D’Ambrosio, E. Mitidieri, and S. I. Pohozaev. Representation formulae and inequalities for solutions of a class of second order partial differential equations.Trans. Amer. Math. Soc., 358(2):893–910, 2006

work page 2006

[15] [15]

Franchi and E

B. Franchi and E. Lanconelli. H¨ older regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10(4):523–541, 1983

work page 1983

[16] [16]

Franchi and E

B. Franchi and E. Lanconelli. Une m´ etrique associ´ ee ` a une classe d’op´ erateurs elliptiques d´ eg´ en´ er´ es.Rend. Sem. Mat. Univ. Politec. Torino, pages 105–114, 1983. Conference on linear partial and pseudodifferential operators (Torino, 1982)

work page 1983

[17] [17]

Franchi and E

B. Franchi and E. Lanconelli. An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality.Comm. Partial Differential Equations, 9(13):1237–1264, 1984

work page 1984

[18] [18]

Garofalo

N. Garofalo. Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension.J. Differential Equations, 104(1):117–146, 1993

work page 1993

[19] [19]

V. V. Grushin. A certain class of hypoelliptic operators.Mat. Sb. (N.S.), 83(125):456– 473, 1970

work page 1970

[20] [20]

C. E. Guti´ errez and E. Lanconelli. Maximum principle, nonhomogeneous Harnack in- equality, and Liouville theorems forX-elliptic operators.Comm. Partial Differential Equations, 28(11-12):1833–1862, 2003

work page 2003

[21] [21]

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

work page 2003

[22] [22]

Hirano, C

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

work page 2004

[23] [23]

Hirano, C

N. Hirano, C. Saccon, and N. Shioji. Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem.J. Differential Equations, 245(8):1997– 2037, 2008. 41

work page 1997

[24] [24]

H¨ ormander

L. H¨ ormander. Hypoelliptic second order differential equations.Acta Math., 119:147–171, 1967

work page 1967

[25] [25]

Jabri.The mountain pass theorem, volume 95 ofEncyclopedia of Mathematics and its Applications

Y. Jabri.The mountain pass theorem, volume 95 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003. Variants, generalizations and some applications

work page 2003

[26] [26]

Lanconelli and A

E. Lanconelli and A. E. Kogoj.X-elliptic operators andX-control distances.Ricerche Mat., 49:223–243, 2000. Contributions in honor of the memory of Ennio De Giorgi (Italian)

work page 2000

[27] [27]

Lazer and J

A. Lazer and J. McKenna. On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc., 111:721–730, 1991

work page 1991

[28] [28]

Leoni.A first course in fractional Sobolev spaces, volume 229 ofGraduate Studies in Mathematics

G. Leoni.A first course in fractional Sobolev spaces, volume 229 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2023

work page 2023

[29] [29]

Loiudice

A. Loiudice. Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators.Ann. Mat. Pura Appl. (4), 198(6):1909– 1930, 2019

work page 1909

[30] [30]

Molica Bisci, P

G. Molica Bisci, P. Malanchini, and S. Secchi. A note on nonlinear critical problems involving the Grushin subelliptic operator: bifurcation and multiplicity results.Potential Anal., 63(3):1059–1072, 2025

work page 2025

[31] [31]

Monti and D

R. Monti and D. Morbidelli. Kelvin transform for Grushin operators and critical semi- linear equations.Duke Math. J., 131(1):167–202, 2006

work page 2006

[32] [32]

I. Shafrir. Asymptotic behaviour of minimizing sequences for Hardy’s inequality.Com- mun. Contemp. Math., 2(2):151–189, 2000

work page 2000

[33] [33]

Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebiete

M. Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Math- ematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial differen- tial equations and H...

work page 2008

[34] [34]

F. G. Tricomi. Sulle equazioni lineari alle derivate parziali di 2 o ordine di tipo misto. Tipografia della R. Accademia nazionale dei Lincei, 14:133–247, 1923

work page 1923

[35] [35]

Zhang and D

J. Zhang and D. Yang. Critical Hardy-Sobolev exponents problem with Grushin operator and Hardy-type singularity terms.Acta Appl. Math., 172:Paper No. 4, 15, 2021

work page 2021

[36] [36]

Zhang and D

J. Zhang and D. Yang. Existence and asymptotic behavior of solution for a degenerate elliptic equation involving Grushin-type operator and critical Sobolev-Hardy exponents. Acta Math. Sci. Ser. A (Chinese Ed.), 41(4):997–1012, 2021. 42

work page 2021