Existence of multiple solutions for quasi-linear degenerate elliptic equations

Yawei Wei

REVIEW 4 major objections 4 minor 15 references

Existence of multiple solutions for quasi-linear degenerate elliptic equations

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Corner-degenerate p-Laplacian: infinitely many weak solutions, and an infinite eigenvalue sequence in the resonant case.

desk verdict The paper targets a real p>2 extension of a corner-degenerate p-Laplacian result, but the weak form it analyzes is not the weak form of the stated strong equation, and the proof of Theorem 1.1 does not cover the full claimed q-range. read the letter →

arxiv 1908.07134 v1 pith:P5Y5PF5V submitted 2019-08-20 math.AP

Yawei Wei This is my paper · ORCID

classification math.AP MSC 35J9235J7035P30

keywords degenerateellipticequationscornersingularityweightedSobolevspacesp-Laplacianvariationalmethodsgenusnonlineareigenvalueproblemmultipleweaksolutions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a quasilinear elliptic equation built from the corner-degenerate gradient operator has infinitely many nontrivial weak solutions. For exponents $20$ the Dirichlet problem has infinitely many nontrivial weak solutions, and their energy levels tend to infinity. For the resonant case $q=p$, the paper states an infinite sequence of eigenpairs $(u_k,\lambda_k)$ with $\lambda_k\to\infty$. If correct, the results show that the degenerate operator inherits the same multiplicity picture as the classical $p$-Laplacian in a corner-geometry setting.

What carries the argument

The central object is the corner-degenerate gradient $\nabla_M=(x_1\partial_{x_1},x_1x_2\partial_{x_2},\partial_{x'})$ with divergence $\mathrm{div}_M=\nabla_M\cdot$, acting on the weighted Sobolev space $H^{1,((N-1)/p,N/p)}_{p,0}(M)$ with measure $d\sigma=x_1^{-2}x_2^{-1}dx_1dx_2dx'$. What carries the argument is the compact embedding of this space into weighted $L^q$ spaces (Lemma 3.1). The even functional is then analyzed by genus of symmetric sets: abstract critical-point lemmas yield infinitely many critical values in the $p<q$ case, while for $q=p$ a tangent flow on the level manifold $\mathcal{M}=\{u:\frac1p\int_M x_1|\nabla_Mu|^p\,d\sigma=\alpha\}$ runs a symmetric min-max scheme with a duality map in place of the gradient.

What would settle it

Compute the Euler-Lagrange equation of the functional $J$ with respect to the measure $d\sigma=x_1^{-2}x_2^{-1}dx_1dx_2dx'$ in ordinary coordinates. The three derivative terms acquire coefficients $x_1x_2^{-1}$, $x_1x_2$, and $x_1^{-1}x_2^{-1}$; equation (1.1) has the single coefficient $(x_1x_2)^{-p}$ in front of $\mathrm{div}_M(\nabla_Mu)$. If the two expressions are not equal, then (1.2) is not the weak form of (1.1), and the infinitely many critical points belong to a different boundary-value problem.

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Extended reading notes

Core claim

On the weighted Sobolev space $H^{1,((N-1)/p,N/p)}_{p,0}(M)$, the energy functional $J(u)=\frac{1}{p}\int_M x_1|\nabla_M u|^p\,d\sigma-\frac{\lambda}{q}\int_M x_1(x_1x_2)^p|u|^q\,d\sigma$ has infinitely many critical points. For $2<p<N$ and $p<q<p^*$, Theorem 1.1 asserts infinitely many nontrivial weak solutions in the sense of (1.2), with critical values $c_m\to\infty$; Theorem 1.3 asserts that when $q=p$ there are infinitely many pairs $(u_k,\lambda_k)$ satisfying the weak equation, with $\lambda_k\to\infty$. The proof uses the compact embedding of the weighted Sobolev space into a weighted Lebesgue space, together with symmetric genus-based min-max arguments.

Load-bearing premise

The load-bearing premise is that the weak equation (1.2) is equivalent to the strong equation (1.1), so the critical points produced are genuine solutions of the boundary-value problem stated in the title.

Editorial extensions

If this is right

For every fixed $\lambda>0$ and $p<q<p^*$, the Dirichlet problem has infinitely many nontrivial weak solutions, with energy levels $c_m\to\infty$.
When $q=p$, there is an infinite sequence of eigenpairs $(u_k,\lambda_k)$ solving the weak equation, with $\lambda_k\to\infty$; the degenerate operator therefore has an unbounded spectrum.
The weighted Sobolev framework transfers the classical $p$-Laplacian multiplicity picture to a model corner domain, so analogous statements should hold on finite stretched corner domains rather than only on $(0,\delta)^2\times X$.
Each weak solution satisfies (1.2), so numerical or regularity analysis can proceed from the variational formulation without treating the strong PDE directly.

Reading between the lines

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

Editorial extension: The paper does not say whether the infinitely many solutions are sign-changing; because the functional is even, applying the same genus count to subsets of fixed sign would be a natural further test, but that conclusion is not in the paper.
Editorial extension: The compact embedding is the main place the corner geometry enters; if the same embedding holds for more general weights or domains, the multiplicity theorem should carry over, though this is not demonstrated.
Editorial extension: If the weak/strong equivalence check described in the falsifier reveals a mismatch, the correct strong equation behind (1.2) can be computed by integrating by parts; the theorems would then describe that corrected problem, not necessarily equation (1.1).

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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

What you should know up front: this paper claims infinitely many solutions for a quasi-linear degenerate p-Laplacian on a stretched corner, but the weak form it actually studies is not the weak form of the stated strong equation. That mismatch alone undermines the central claim as written. There is also an independent gap in the proof of Theorem 1.1: the variational argument only covers q < p(p+1), not the full stated range p < q < p*.

The problem itself is new and reasonable. The p>2 extension of the p=2 case from [5] is a legitimate question, and the corner-degenerate weighted Sobolev setup is appropriate. The variational machinery — Ambrosetti–Rabinowitz for p

The soft spots are serious. First, equation (1.1) has -(x1x2)^{-p} div_M(...) on the left; integrating by parts against dσ = dx1/x1 dx2/(x1x2) dx' gives a term with weight (x1x2)^{-p}, not x1. The functional J defines critical points for (1.2), not for (1.1). So the theorems solve a different equation. This is not a one-line typo; the weights propagate through the entire argument. Second, Proposition 3.5 asserts q < p* < p(p+1), but that inequality requires N > p+1. For N ≤ p+1, the theorem's range includes q ≥ p(p+1), where the embedding in Lemma 3.1 fails and J may not even be well-defined. The stress-test note is correct about this. Third, Proposition 3.6 is simply omitted, though it is needed for condition I5. Fourth, Lemma 4.1 uses a finite binomial expansion for real p, which is invalid; the map z ↦ |z|^{p-2}z needs a mean-value or Hölder-type estimate instead.

None of this is fatal to the underlying idea, but as written the theorems are not proved. I would not cite this version. A serious referee could still be useful, because the author has identified a real problem and the gaps are identifiable and likely fixable, but the paper needs substantial revision before it can be accepted.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the quasi-linear degenerate elliptic Dirichlet problem (1.1) on a model corner manifold M=(0,δ)^2×X, where the degenerate operator is built from ∇M=(x1∂x1, x1x2∂x2, ∂x') and divM=∇M·. The paper defines weak solutions through (1.2) and an energy functional J, then claims in Theorem 1.1 that for 2<p<N and p<q<p*, λ>0, there are infinitely many nontrivial weak solutions, with critical values tending to infinity in Theorem 1.2. For the case p=q, Theorems 1.3 and 1.4 claim an infinite sequence of eigenvalue-eigenvector pairs (u_k, λ_k) with λ_k→∞. Section 3 uses abstract critical-point theorems from [2], establishing (PS) and conditions (I1)–(I5), while Section 4 employs a Lusternik-Schnirelman argument on a constraint manifold.

Significance. If the results were correct, they would extend known multiplicity theorems for the p-Laplacian to a class of corner-degenerate operators, which is a genuinely useful direction given the applications to corner singularities. The paper has clear strengths: it works in a weighted Sobolev framework, gives a detailed (PS) verification using a Brezis-Lieb-type lemma, and follows explicit abstract minimax schemes for both the superlinear and the eigenvalue cases. However, several load-bearing gaps prevent the claims from being accepted as proved: the weak form does not correspond to the stated PDE under the paper's own definitions, the q-range in Theorem 1.1 is not covered by the embedding used, a required condition (I5) is left unproved, and the p=q argument relies on an invalid binomial expansion for non-integer p.

major comments (4)

[§1, Eq. (1.1) vs. Eq. (1.2)] The weak form (1.2) is not equivalent to the PDE (1.1) under the definitions given in the introduction. With dσ=dx1/x1 dx2/(x1x2)dx', direct integration by parts converts the left side of (1.2) into −∫ φ x1x2[∂_{x1}(A_1 x2^{-1}) + ∂_{x2}(A_2 x1) + ∂_{x'}(A' (x1x2)^{-1})] dσ, where A=|∇_M u|^{p−2}∇_M u. The strong operator in (1.1) is (x1x2)^{−p}div_M A = (x1x2)^{−p}(x1∂_{x1}A_1+x1x2∂_{x2}A_2+∂_{x'}A'), which has different weights and different derivative structure. Since the paper neither proves nor derives the equivalence asserted in the sentence after (1.2), the theorems prove existence of solutions to a different equation than the one stated.
[§3.1, Proposition 3.5 and Lemma 3.1] Theorem 1.1 claims p<q<p*, but Proposition 3.5 uses the condition q<p*<p(p+1). This condition holds only when N>p+1; for 2<p<N with N≤p+1 one has p*≥p(p+1), so the theorem's range includes q≥p(p+1). The embedding in Lemma 3.1 with the weights γ'=((N−p−1)/q,(N−p)/q) used in the energy estimates requires (p+1)/q>1/p, i.e. q<p(p+1). For q≥p(p+1) the nonlinear term ∫ x1^{p+1}x2^p|u|^q dσ need not be finite on H^{1,((N−1)/p,N/p)}_{p,0}(M), so J is not defined on the claimed solution space. The proof as written therefore covers only q<p(p+1), not the full range stated in Theorem 1.1.
[§3.1, Proposition 3.6] Condition (I5) is a hypothesis of Lemmas 2.9 and 2.10 and is verified by Proposition 3.6, but the proposition is not proved; the text says only "We omit the easy proof of Proposition 3.6 here for the limit length of writing." Since the conclusion of Theorem 1.1 depends on these lemmas, an omitted proof of a required condition leaves a gap in the derivation of the critical-value sequence.
[§4.1, Eq. (4.3)] The proof of Lemma 4.1 applies the binomial theorem to |u0+δ|^{p−2}(u0+δ) and expands it in powers of δ up to p−2. Here p is a real number in (2,N), not an integer, so the finite binomial expansion with combinatorial coefficients is not valid. The same invalid expansion is used in Lemma 4.2 and in the estimates for D(u). Since uniform continuity and compactness of b underpin the deformation and minimax argument, the proof of Theorem 1.3 is not supported.

minor comments (4)

[Abstract and Theorems 1.1, 1.3] There are typos: "degenera te" in the abstract and "processes" should be "possesses" in Theorems 1.1 and 1.3.
[§2, Proposition 2.4] The proof of Proposition 2.4 simply says "Follow the same process of Proposition 3.2 in [5]"; at a minimum the relevant hypotheses and the role of the cut-off functions should be stated so the Poincaré-type inequality is verifiable from the text.
[§3.1, Lemma 3.4] In the (PS) verification, the line J(u_k)−(1/q)<J'(u_k),u_k> uses notation that is not fully defined; in particular <J'(u_k),u_k> should be the pairing between H^{-1,(-(N−1)/p,−N/p)}_p and H^{1,((N−1)/p,N/p)}_{p,0}(M), and the displayed formula for the difference <J'(u_k)−J'(u),u_k−u> should include the weight x1 explicitly for consistency.
[§4.1, Lemma 4.8] In Lemma 4.8 the dual space is written H^{-1,(-N−1/p,−N/p)}_p(B), where B is not defined in this subsection; this appears to be a relic of the earlier notation for the unit ball and should be M.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity; self-citations supply only the weighted-space framework, not the existence conclusion.

full rationale

The paper proves two existence theorems for a degenerate p-Laplacian-type equation. There are no fitted parameters, no data subsets, and no predictions that could be forced by construction, so the empirical-circularity patterns do not apply. The derivation chain is: define weighted corner Sobolev spaces (following Schulze [14] and Schulze-Wei [15]), import a Poincaré inequality (Prop. 2.4: 'Follow the same process of Proposition 3.2 in [5]'), prove a weighted embedding (Lemma 3.1), and then verify the hypotheses of abstract mountain-pass/genus theorems ([2], [1], [12], [13]) for the functional J. The self-citations [5], [14], [15] supply the corner-degenerate functional-analytic framework, but the load-bearing variational mechanism (PS condition, genus estimates, deformation flow) is carried out in the paper itself against external abstract critical-point theorems. A cited framework is independent support here because the cited results are embedding and norm-equivalence tools and do not assert infinitely many solutions of (1.1). The paper does contain serious proof gaps: Proposition 3.6 is omitted ('We omit the easy proof of Proposition 3.6 here'), Proposition 3.5 uses the inequality q < p* < p(p+1), which fails when N = p+1, and Lemma 4.1 applies a finite binomial expansion for a real exponent p. These are correctness risks, not circularity: none of them makes the conclusion identical to an input by definition. The asserted equivalence of (1.1) with the weak form (1.2) is also a substantive analytic claim; even if the integration by parts is incorrect, that would make the theorems address a different equation, not make the derivation circular. Overall, no step reduces to its own input; the only noteworthy feature is non-load-bearing reliance on the author's prior weighted-space framework, so the score is 2 rather than 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. It relies on the weighted Sobolev space framework from Schulze's corner calculus [14] and the Poincaré inequality from [5], as well as standard abstract critical point theorems. The unverified assumption that (1.1) and (1.2) are equivalent is the main additional burden.

assumptions (4)

domain assumption The weighted Sobolev spaces H^{m,(γ1,γ2)}_{p,0}(M) and their compact embedding properties (Lemma 3.1) are valid.
The framework is taken from Schulze's corner calculus [14] and the compact embedding proof in Lemma 3.1 has several notational gaps and relies on the author's prior work [5].
domain assumption Poincaré inequality (Proposition 2.4) holds for the weighted norm ‖∇_M·‖_{L^{γ1,γ2}_p}.
The proof is 'Follow the same process of Proposition 3.2 in [5]', so it is not reproduced.
standard math The abstract critical point theorems (Lemma 2.9 and 2.10) apply to J.
These are cited from [2] and are standard, but the paper must verify conditions (I1)-(I5); condition (I5) is not proved (Proposition 3.6 omitted).
ad hoc to paper Binomial expansion can be applied to the real power |u0+δ|^{p-2}(u0+δ).
Used in Lemma 4.1 and 4.2 with binomial coefficients C_l^{p-2} as if p-2 were an integer; p is real, so the step is unjustified.

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Cite this review

Pith. "Pith review of Existence of multiple solutions for quasi-linear degenerate elliptic equations." pith.science (2026). https://pith.science/paper/P5Y5PF5V

@misc{pith2026190807134,
  author       = {Pith},
  title        = {Pith review of: Existence of multiple solutions for quasi-linear degenerate elliptic equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P5Y5PF5V}},
  note         = {Machine review of arXiv:1908.07134}
}

read the original abstract

The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity many solutions for the problems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

H. Chen, X. Liu and Y. Wei,Multiple solutions for semi-li near corner degenerate elliptic equations, Journal of Functional Analysis 266(2014)3815¨ C3839

work page 2014

[2] [2]

Ambrosetti, P

A. Ambrosetti, P. H. Rabinowitz, Dual variational metho ds in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-3 81

work page 1973

[3] [1]

Amann, Lusternik-Schnirelman Theory and Nonlinear e igenvalue problems Math

H. Amann, Lusternik-Schnirelman Theory and Nonlinear e igenvalue problems Math. Ann. 199 (1972), 55-72

work page 1972

[4] [3]

Brezis, Functional Analysis, Sobolev spaces, and Par tial Diﬀerential equations, Springer, 2011

H. Brezis, Functional Analysis, Sobolev spaces, and Par tial Diﬀerential equations, Springer, 2011

work page 2011

[5] [4]

Carl , D

S. Carl , D. Motreanu, Multiple and sign-changing soluti ons for the multivalued p -Laplacian equation Mathematische Nachrichten, 2010 , 283 (7) :965-981

work page 2010

[6] [6]

M. DAUGE. Elliptic Boundary Value Problems in Corner Dom ains ¨C Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, Vo l. 1341. Springer- Verlag, Berlin 1988

work page 1988

[7] [7]

Drabek, Resonance Problems for the p -Laplacian, Jour nal of Functional Analysis, 1999 , 169 (1) :189-200

P. Drabek, Resonance Problems for the p -Laplacian, Jour nal of Functional Analysis, 1999 , 169 (1) :189-200

work page 1999

[8] [8]

Garcia Azorero, I

J. Garcia Azorero, I. Peral Alonso, Existence and nonuni queness for the p-Laplacian: Nonlinear Eigenvalues, Commun. in PDE 12(1987), 1389-1430

work page 1987