Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator

Giovanni Molica Bisci; Paolo Malanchini; Simone Secchi

REVIEW 2 major objections 4 minor 2 cited by

Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator

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Pith's one-line read This paper proves that the critical Dirichlet problem for the p-Grushin operator has nontrivial solution pairs branching off every eigenvalue of a cohomological-index spectrum, with the number of branches equal to the eigenvalue…

desk verdict A genuine p>1 extension of the p=2 Grushin multiplicity theorem with a correct-looking strategy, but Lemma 3.2 is false as stated and the (PS) proof leans on it; the gap looks repairable, so referee it. read the letter →

arxiv 2501.11013 v1 pith:C2Q2SPU6 submitted 2025-01-19 math.AP

Paolo Malanchini , Giovanni Molica Bisci , Simone Secchi This is my paper · ORCID

classification math.AP MSC 35J9235J7035B3235B3335P30

keywords p-GrushinoperatorcriticalSobolevexponentbifurcationmultiplicitycohomologicalindexconcentration-compactnessdegeneratequasilinearellipticcompactnesscondition

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

For a bounded domain in $\mathbb{R}^m\times\mathbb{R}^\ell$, this paper studies the critical Dirichlet problem $-\Delta_\gamma^p u=\lambda|u|^{p-2}u+|u|^{p_\gamma^*-2}u$ for the degenerate p-Grushin operator and the critical Sobolev exponent $p_\gamma^*$. It proves that nontrivial solution pairs exist in every interval $(\lambda_k - S/|\Omega|^{p/N_\gamma},\lambda_k)$ of an eigenvalue sequence, and that when $\lambda_{k+1}$ is repeated $m$ times and $\lambda$ lies just below it, the problem has $m$ distinct pairs of solutions converging to zero as $\lambda\uparrow\lambda_{k+1}$. This extends to all $p>1$ a bifurcation and multiplicity result previously available only for $p=2$, where linear eigenspaces exist; for $p\neq2$ the nonexistence of eigenspaces is bypassed with an abstract critical-point theorem based on the $\mathbb{Z}_2$-cohomological index. The proof also supplies a concentration-compactness principle tailored to the p-Grushin operator.

What carries the argument

The load-bearing objects are the p-Grushin operator $\Delta_\gamma^p u=\nabla_\gamma\cdot(|\nabla_\gamma u|^{p-2}\nabla_\gamma u)$ with degenerate gradient $\nabla_\gamma=(\nabla_x,|x|^\gamma\nabla_y)$, the homogeneous dimension $N_\gamma=m+(1+\gamma)\ell$, the critical exponent $p_\gamma^*=pN_\gamma/(N_\gamma-p)$, and the best Sobolev constant $S$. The argument runs on three pieces: a minimax eigenvalue sequence $\{\lambda_k\}$ defined by requiring the $\mathbb{Z}_2$-cohomological index of a symmetric set on the unit sphere to be at least $k$; an abstract pseudo-index critical-point theorem that turns a separation between index levels $k$ and $k+m$ into $m$ distinct pairs of critical points; and a concentration-compactness argument that proves the needed compactness condition below the threshold $S^{N_\gamma/p}/N_\gamma$. The numerical condition (1.5) comes from the one-variable maximization $\sup_{\rho\ge0}\big[(\lambda_{k+1}-\lambda)\rho^p/p-\rho^{p_\gamma^*/p}/(p_\gamma^*|\Omega|^{p_\gamma^*/N_\gamma})\big]=|\Omega|(\lambda_{k+1}-\lambda)^{N_\gamma/p}/N_\gamma$, with $\rho=\int_\Omega|u|^p\,dz$.

What would settle it

Take $\mu=\nu$ equal to Lebesgue measure on a bounded domain with $p=q$ and $C_0=1$: the inequality in Lemma 3.2 holds, yet $\nu$ has no atoms, so the lemma's purely atomic conclusion fails; consequently the proof of Theorem 3.3 must be modified to keep an absolutely continuous term $|u|^{p_\gamma^*}dz$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: the boundary-value problem (1.3) bifurcates at every eigenvalue of the p-Grushin operator, in the sense that each interval just below an eigenvalue carries nontrivial solution pairs, and a repeated eigenvalue of multiplicity $m$ generates $m$ distinct pairs $\pm u_j^\lambda$ with $u_j^\lambda\to 0$ in $\mathring{W}^{1,p}_\gamma(\Omega)$ as $\lambda\uparrow\lambda_{k+1}$. The precise window is $\lambda_k\le\lambda<\lambda_{k+1}=\cdots=\lambda_{k+m}<\lambda_{k+m+1}$ with $\lambda>\lambda_{k+1}-S/|\Omega|^{p/N_\gamma}$. Because $-\Delta_\gamma^p$ is not linear for $p\neq2$, the classical linking argument on eigenspaces is replaced by a pseudo-index critical-point theorem and a concentration-compactness principle for the Grushin setting, and the solution branches are shown to vanish in the ambient norm as the parameter reaches the eigenvalue.

Load-bearing premise

The load-bearing premise is the assertion of Lemma 3.2 that a measure satisfying a Sobolev-type inequality against all test functions must be purely atomic, which is false in general and leaves the concentration-compactness step and the required compactness condition without a valid proof as written.

Editorial extensions

If this is right

Corollary 1.2 gives a nontrivial solution of (1.3) for every $\lambda\in\bigcup_{k\ge1}(\lambda_k-S/|\Omega|^{p/N_\gamma},\lambda_k)$.
At a repeated eigenvalue $\lambda_{k+1}$ of multiplicity $m$, the problem admits $m$ distinct pairs of nontrivial solutions, so the number of branches matches the multiplicity.
Each branch $\pm u_j^\lambda$ vanishes in the $\mathring{W}^{1,p}_\gamma(\Omega)$ norm as $\lambda$ approaches $\lambda_{k+1}$, so every eigenvalue is a bifurcation point in the sense of nontrivial solutions appearing from zero.
The theorem covers every $p>1$, including the range where the operator has no linear eigenspaces and classical linking arguments fail.

Reading between the lines

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

Beyond the paper, the gap in Lemma 3.2 is likely repairable: a standard concentration-compactness argument keeps the absolutely continuous part $|u|^{p_\gamma^*}dz$ separate from the atomic peaks, and with that correction the estimates of Lemma 6.1 appear to go through.
Beyond the paper, the same cohomological-index-plus-pseudo-index scheme should transfer to other degenerate quasilinear critical problems, provided one can verify a compactness threshold below $S^{N_\gamma/p}/N_\gamma$.
Beyond the paper, condition (1.5) uses the domain's measure and the universal constant $S$; a sharper shape-dependent constant could widen the intervals on which bifurcation is proved.
Beyond the paper, the paper leaves open the asymptotic profile of the branches near $\lambda_{k+1}$; a natural next step is to compute the rate of vanishing and the concentration behavior of $u_j^\lambda$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: genuinely new main theorem, standard but sound strategy, and one real gap that looks repairable. Referee it, expect major revision, but don't cite the current version.

What is new: Theorem 1.1 extends the p=2 Grushin bifurcation/multiplicity result of the same authors to general p>1, exactly where the operator has no linear eigenspaces. The pseudo-index machinery from Perera–Squassina–Yang [21] and the Z2-cohomological eigenvalue sequence from [19] are assembled cleanly for the p-Grushin operator, and the paper writes out a concentration-compactness principle (Theorem 3.3) for this operator that will be a useful standalone tool. The (PS) verification, the sublevel estimates, and the final bifurcation argument are routine in outline but carefully executed, and I read them as correct modulo the gap below.

The load-bearing problem is Lemma 3.2. As stated it claims that the inequality (∫|φ|^q dν)^{1/q} ≤ C0 (∫|φ|^p dμ)^{1/p} with 1 ≤ p ≤ q forces ν to be purely atomic. That is false — take p=q and μ=ν=Lebesgue measure. Lemma 3.2 is then used in Theorem 3.3 to identify the diffuse part of the critical measure, and Lemma 6.1 proves the (PS) condition through Theorem 3.3, so the main theorem is not rigorously proved from the printed text. The stress test holds up on reading. That said, the fix looks standard: the applications only need q=p*_γ>p, where the correct Lions lemma ([16]) gives exactly the decomposition the authors need, and their own Brezis–Lieb split into a residual measure already points that way. I am fairly confident the theorem survives with Lemma 3.2 restricted to p

Minor points: the abstract announces a new abstract result, but Theorem 5.1 is quoted from [21, Theorem 2.2] — fine to use, just not new. There is a typo in the proof of Theorem 3.3, '|∇γvn|2 dz' should be p. The citation pattern is honest; self-citation [2] is only motivational.

This is a paper for the subfield of subelliptic quasilinear critical problems: people will want the concentration-compactness principle and the multiplicity result. It changes nothing global, but it settles a natural open case. Recommendation: send to a serious referee focused on Section 3; expect conditional acceptance once the lemma is repaired and the abstract corrected. As printed I would not cite it; after the fix I would.

Referee Report

2 major / 4 minor

Summary. The paper proves a bifurcation and multiplicity theorem for the critical p-Grushin problem (1.3) for general p>1, extending the p=2 result of [2]. The proof combines a concentration-compactness principle for the p-Grushin operator (Section 3), a variational spectrum based on the Z2-cohomological index (Section 4), an abstract critical point theorem with pseudo-index (Section 5), and a Palais-Smale verification below the critical threshold (Section 6). Theorem 1.1 asserts that if λ lies near an eigenvalue λ_{k+1} from below and above λ_{k+1} - S/|Ω|^{p/Nγ}, then there are as many distinct pairs of solutions as the multiplicity of λ_{k+1}.

Significance. The result is a natural and nontrivial extension to p≠2 of known bifurcation results; because the p-Grushin operator has no linear eigenspaces for p≠2, the use of the cohomological index and pseudo-index is appropriate. The paper also provides a version of Lions' concentration-compactness principle for this degenerate operator. The structural chain of the proof is standard, and the abstract theorem is cited from [21]. However, the concentration-compactness step as written relies on a lemma that is false in the stated generality, so the central claim is not rigorously established in the current text. Since the false case p=q is not used in the applications (q=p*_γ>p), the gap appears repairable.

major comments (2)

[Section 3, Lemma 3.2] Lemma 3.2 is false as stated. For p=q=2, take μ=ν to be Lebesgue measure on a bounded domain and C0=1; the inequality (3.5) is an identity, yet ν is not a sum of Dirac masses. The proof of Theorem 3.3 applies this lemma both in the case u≡0 and to the residual measures ω and λ in the general case, and Theorem 3.3 is used in Lemma 6.1 to prove the Palais-Smale condition. Thus the proof of Theorem 1.1 as written contains a gap at a load-bearing point. Because the applications always have q=p*_γ>p, the lemma is valid in that regime, so the gap is repairable by restricting Lemma 3.2 to 1≤p<q and verifying the precise statement from [16]. This correction must be made before the stated results are proved.
[Theorem 1.1 and Lemma 6.1] The Palais-Smale condition is the only place where the concentration-compactness theorem enters the proof of Theorem 1.1. Since Lemma 3.2 is false as stated, the current proof of the (PS) condition is incomplete. The authors should either prove the corrected version of Lemma 3.2 (with p<q) or give a direct argument ruling out the diffuse part of the residual measure in the specific setting of the Grushin operator. This is not an external counterexample to the theorem; it is a necessary repair to the proof.

minor comments (4)

[Section 3, proof of Theorem 3.3] The sentence 'That is holds (3.5)... So Theorem 3.3 is proved in the case u ≡ 0 applying Lemma 3.2' contains a grammatical error and should read 'That is, (3.5) holds... So (3.2) is proved in the case u ≡ 0 by Lemma 3.2.'
[Section 3, proof of Theorem 3.3] In the definition of λ_n after (3.6), the text writes λ_n = |∇γ v_n|^2 dz; this should be |∇γ v_n|^p dz to match the p-norm used elsewhere.
[Section 4] In the definition of sublevels and superlevels, the line 'Ψ^a = {u ∈ ˚W | Φ(u) ≤ a}' should presumably be 'Ψ^a = {u ∈ ˚W | Ψ(u) ≤ a}', since Φ is not otherwise defined there. Also, in Proposition 4.2 and Proposition 6.4, the sets Ψ_a and Ψ^a are used as subsets of M; this should be stated explicitly to avoid ambiguity.
[Section 6, proof of Lemma 6.1] In the estimate for Ap(un)-Ap(um), the constant c2 is written as depending on M = max{||un||_{γ,p}, ||um||_{γ,p}}; for clarity, the displayed formula should indicate that c2 depends on M and on N, γ, p, as stated in the preceding sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained and rests on external abstract theorems, with no fitted parameters renamed as predictions.

full rationale

The proof chain of Theorem 1.1 is an application of external results: the abstract critical point theorem comes from [21, Theorem 2.2], the eigenvalue sequence defined by the Z2-cohomological index comes from [19, Theorem 4.6], and the concentration-compactness lemma is quoted from Lions [16, Lemma 1.2 and Remark 1.5]. The condition (1.5) and the Palais-Smale level b = S^{N_gamma/p}/N_gamma are computed directly from the optimal Sobolev constant S, the domain measure, and the eigenvalue lambda_{k+1}; there are no fitted parameters and no quantity being predicted is built into the hypotheses. The authors' own prior work [2] is cited only as background for the p=2 case, and the text explicitly says the eigenspace linking argument from [2] is not used because the nonlinear operator has no linear eigenspaces when p != 2; hence that self-citation is not load-bearing. The reader-identified issue with Lemma 3.2 (the assertion that the measure nu is purely atomic is false in the p = q case, e.g. mu = nu = Lebesgue measure) is a correctness gap in the concentration-compactness argument, not a circularity: the lemma is stated as coming from an external source and is not an input assumed to prove itself. Thus no step reduces by definition or by self-citation to its own conclusion.

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

No free parameters or invented entities. The proof rests on a chain of external results (Sobolev embedding, Perera's spectral theory, the pseudo-index theorem of [21]) plus the paper's new concentration-compactness argument, whose key lemma is questionable.

assumptions (6)

standard math Sobolev embedding for the p-Grushin space: the space W^{1,p}_gamma(Omega) embeds compactly into L^q(Omega) for q < p*_gamma = pN_gamma/(N_gamma-p), and continuously at q = p*_gamma.
Quoted from [13, Proposition 3.2, Theorem 3.3] and [11, Corollary 2.11]; used to define the critical exponent and best constant S in (2.1)-(2.2), and in the concentration-compactness argument.
standard math The operator A_p(u) = -div_gamma(|nabla_gamma u|^{p-2} nabla_gamma u) satisfies properties (A1)-(A4), including the (S)+ property and strict monotonicity.
Invoked in Lemma 6.1 (strong convergence from Palais-Smale) and Lemma 6.2 (strict monotonicity). The (S)+ property is imported from [19, Proposition 1.3]; strict monotonicity from [19, Lemma 6.3].
standard math The cohomological index construction yields a nondecreasing sequence {lambda_k} of eigenvalues of -Delta_gamma^p with the properties (1)-(3) of Proposition 4.2.
Imported from [19, Theorem 4.6] and [18,22]. This eigenvalue sequence is what Theorem 1.1 refers to.
standard math The abstract pseudo-index critical point theorem (Theorem 5.1) is valid.
Quoted from [21, Theorem 2.2] (proof not reproduced). Despite the abstract's claim, the paper does not prove a new abstract theorem.
ad hoc to paper Lemma 3.2's atomic-only conclusion for measures satisfying (3.5) is valid.
This is the fragile premise: the quoted statement is false for general measures (counterexample with mu=nu=Lebesgue and p=q), and the proof of Theorem 3.3 needs a correct version including absolutely continuous parts.
domain assumption The homogeneous dimension N_gamma = m + (1+gamma)ell is the right dimension for the critical exponent.
Standard for Grushin operators; defines p*_gamma.

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Pith. "Pith review of Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator." pith.science (2026). https://pith.science/paper/C2Q2SPU6

@misc{pith2026250111013,
  author       = {Pith},
  title        = {Pith review of: Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2Q2SPU6}},
  note         = {Machine review of arXiv:2501.11013}
}

abstract

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator $\Delta_\gamma^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the nonlinear operator $-\Delta_\gamma^p$ has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem which is not based on linear subspaces. We also prove a new abstract result based on a pseudo-index related to the $\mathbf{Z}_2$-cohomological index that is applicable here. We provide a version of the Lions' Concentration-Compactness Principle for our operator.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 2 Pith papers

[21]

Bifurc ation and multi- plicity results for critical p-Laplacian problems

Kanishka Perera, Marco Squassina, and Yang Yang. Bifurc ation and multi- plicity results for critical p-Laplacian problems. Topol. Methods Nonlinear Anal., 47(1):187–194, 2016

work page 2016
[19]

Agarwal, and Donal O’Regan

Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan. Morse theoretic aspects of p-Laplacian type operators, volume 161 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010

work page 2010
[16]

P.-L. Lions. The concentration-compactness principl e in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985

work page 1985
[2]

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

Giovanni Molica Bisci, Paolo Malanchini, and Simone Secc hi. A note on nonlinear critical problems involving the Grushin Subelli ptic Operator: bi- furcation and multiplicity results. arXiv preprint arXiv:2402.17476 , 2024

work page Pith review arXiv 2024
[1]

Alves, Somnath Gandal, Annunziata Loiudi ce, and Jagmo- han Tyagi

Claudianor O. Alves, Somnath Gandal, Annunziata Loiudi ce, and Jagmo- han Tyagi. A Brézis-Nirenberg type problem for a class of dege nerate ellip- tic problems involving the Grushin operator. J. Geom. Anal. , 34(2):Paper No. 52, 41, 2024

work page 2024
[3]

A relation between pointwis e convergence of functions and convergence of functionals

Haïm Brézis and Elliott Lieb. A relation between pointwis e convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. , 88(3):486–490, 1983

work page 1983
[4]

Positive solutions of no nlinear elliptic equations involving critical Sobolev exponents

Haïm Brézis and Louis Nirenberg. Positive solutions of no nlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. , 36(4):437–477, 1983

work page 1983
[5]

Bifurcation and multiplicity results for nonlinear elliptic problems invo lving critical Sobolev exponents

Giovanna Cerami, Donato Fortunato, and Michael Struwe. Bifurcation and multiplicity results for nonlinear elliptic problems invo lving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire , 1(5):341–350, 1984. 21

work page 1984

Show all 26 references

[6]

Ju. V. Egorov. Subelliptic operators. Uspehi Mat. Nauk, 30(2(182)):57–114, 1975

1975
[7]

Fadell and Paul H

Edward R. Fadell and Paul H. Rabinowitz. Generalized coh omological index theories for Lie group actions with an application to b ifurcation questions for Hamiltonian systems. Invent. Math. , 45(2):139–174, 1978

1978
[8]

Multiplicity results for fractional Laplace problems with critical grow th

Alessio Fiscella, Giovanni Molica Bisci, and Raﬀaella Se rvadei. Multiplicity results for fractional Laplace problems with critical grow th. Manuscripta Math., 155(3-4):369–388, 2018

2018
[9]

Modern methods in the calculus of variations: Lp spaces

Irene Fonseca and Giovanni Leoni. Modern methods in the calculus of variations: Lp spaces. Springer Monographs in Mathematics. Springer, New York, 2007

2007
[10]

Quasilinear ellipt ic equations involving critical Sobolev exponents

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

1989
[11]

P -Laplace equation for the Grushin type operator

Ji Zheng Huang and Xin Xin Yang. P -Laplace equation for the Grushin type operator. Acta Math. Sin. (Engl. Ser.) , 39(5):923–938, 2023

2023
[12]

Lq regularity for p-Laplace type Baouendi-Grushin equations

Xiaotao Huang, Feiyao Ma, and Lihe Wang. Lq regularity for p-Laplace type Baouendi-Grushin equations. Nonlinear Anal., 113:137–146, 2015

2015
[13]

Kogoj and Ermanno Lanconelli

Alessia E. Kogoj and Ermanno Lanconelli. On semilinear ∆λ-Laplace equa- tion. Nonlinear Analysis: Theory, Methods & Applications , 75(12):4637– 4649, 2012

2012
[14]

On the equation div (|∇u|p−2∇u) + λ|u|p−2u = 0

Peter Lindqvist. On the equation div (|∇u|p−2∇u) + λ|u|p−2u = 0 . Proc. Amer. Math. Soc. , 109(1):157–164, 1990

1990
[15]

Addendum: On the equation div (|∇u|p−2∇u) + λ|u|p−2u = 0 [Proc

Peter Lindqvist. Addendum: On the equation div (|∇u|p−2∇u) + λ|u|p−2u = 0 [Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164; MR1007505 (90h:35088)]. Proc. Amer. Math. Soc. , 116(2):583–584, 1992

1990
[17]

P.-L. Lions. The concentration-compactness principl e in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana, 1(2):45–121, 1985. 22

1985
[18]

Nontrivial critical groups in p-Laplacian problems via the Yang index

Kanishka Perera. Nontrivial critical groups in p-Laplacian problems via the Yang index. Topol. Methods Nonlinear Anal. , 21(2):301–309, 2003

2003
[20]

Bifurc ation and mul- tiplicity results for critical fractional p-Laplacian problems

Kanishka Perera, Marco Squassina, and Yang Yang. Bifurc ation and mul- tiplicity results for critical fractional p-Laplacian problems. Math. Nachr. , 289(2-3):332–342, 2016

2016
[22]

p-Laplacian problems where the nonlinearity crosses an eigenvalue

Kanishka Perera and Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete Contin. Dyn. Syst. , 13(3):743– 753, 2005

2005
[23]

Régularité de la solution d’une équatio n non linéaire dans RN

Jacques Simon. Régularité de la solution d’une équatio n non linéaire dans RN . In Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 19 77), volume 665 of Lecture Notes in Math. , pages 205–227. Springer, Berlin, 1978

1978
[24]

Variational methods, volume 34 of Ergebnisse der Math- ematik und ihrer Grenzgebiete

Michael Struwe. Variational methods, volume 34 of Ergebnisse der Math- ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern S urveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Se- ries of Modern Surveys in Mathematics] . Springer-Verlag, Ber...

2008
[25]

Liouville- type theorem for nonlinear elliptic equations involving p-Laplace-type Grushin operators

Yunfeng Wei, Caisheng Chen, Qiang Chen, and Hongwei Yan g. Liouville- type theorem for nonlinear elliptic equations involving p-Laplace-type Grushin operators. Math. Methods Appl. Sci. , 43(1):320–333, 2020

2020
[26]

Nontrivial solut ions for an asymp- totically linear ∆α-Laplace equation

Jiafa Xu, Jia Chen, and Donal O’Regan. Nontrivial solut ions for an asymp- totically linear ∆α-Laplace equation. Nonlinear Anal. Model. Control , 28(5):841–858, 2023. 23 Paolo Malanchini, Dipartimento di Matematica e Applicazio ni, Università degli Studi di Milano Bicocca, via...

2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [21]

Bifurc ation and multi- plicity results for critical p-Laplacian problems

Kanishka Perera, Marco Squassina, and Yang Yang. Bifurc ation and multi- plicity results for critical p-Laplacian problems. Topol. Methods Nonlinear Anal., 47(1):187–194, 2016

work page 2016

[2] [19]

Agarwal, and Donal O’Regan

Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan. Morse theoretic aspects of p-Laplacian type operators, volume 161 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010

work page 2010

[3] [16]

P.-L. Lions. The concentration-compactness principl e in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985

work page 1985

[4] [2]

A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

Giovanni Molica Bisci, Paolo Malanchini, and Simone Secc hi. A note on nonlinear critical problems involving the Grushin Subelli ptic Operator: bi- furcation and multiplicity results. arXiv preprint arXiv:2402.17476 , 2024

work page Pith review arXiv 2024

[5] [1]

Alves, Somnath Gandal, Annunziata Loiudi ce, and Jagmo- han Tyagi

Claudianor O. Alves, Somnath Gandal, Annunziata Loiudi ce, and Jagmo- han Tyagi. A Brézis-Nirenberg type problem for a class of dege nerate ellip- tic problems involving the Grushin operator. J. Geom. Anal. , 34(2):Paper No. 52, 41, 2024

work page 2024

[6] [3]

A relation between pointwis e convergence of functions and convergence of functionals

Haïm Brézis and Elliott Lieb. A relation between pointwis e convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. , 88(3):486–490, 1983

work page 1983

[7] [4]

Positive solutions of no nlinear elliptic equations involving critical Sobolev exponents

Haïm Brézis and Louis Nirenberg. Positive solutions of no nlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. , 36(4):437–477, 1983

work page 1983

[8] [5]

Bifurcation and multiplicity results for nonlinear elliptic problems invo lving critical Sobolev exponents

Giovanna Cerami, Donato Fortunato, and Michael Struwe. Bifurcation and multiplicity results for nonlinear elliptic problems invo lving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire , 1(5):341–350, 1984. 21

work page 1984