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On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities

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Pith's one-line read A two-term nonlinear p-Laplace equation is shown to be equivalent to the critical point equation of a single one-parameter Rayleigh quotient, transferring existence and multiplicity results between the two descriptions.

desk verdict A careful, systematic paper that turns a one-line Euler-Lagrange identity into a full dictionary between solutions of p-Laplace equations with polynomial nonlinearities and critical points of a two-parameter Rayleigh quotient; the degenerate-solution characterization is the real payoff. read the letter →

arxiv 2511.10199 v1 pith:6ZDGNYKB submitted 2025-11-13 math.AP

Vladimir Bobkov , Mieko Tanaka This is my paper · ORCID
classification math.AP MSC 35J9235B3035A0135B3835P3047J1049J3549R05
keywords p-LaplaciangeneralizedRayleighquotientconvex-concavesubhomogeneoussuperhomogeneousvariationaleigenvaluesLusternik–Schnirelmannlevelsdegeneratesolutions
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

The paper's central claim is that the two-parameter-looking family of equations −Δₚu = μ|u|^{q−2}u + |u|^{r−2}u is actually governed by a single parameter: its solutions are, up to scaling, the critical points of the homogeneous Rayleigh quotient R_α(u) = ‖∇u‖ₚ^p/(‖u‖_q^{αp}‖u‖_r^{p−αp}) indexed by α. This dictionary lets one treat the linear parameter μ and the quotient parameter α as the same object, so existence, multiplicity, and qualitative properties can be transferred between the two descriptions. The paper applies this to the convex-concave case q

What carries the argument

The central object is R_α(u) = ‖∇u‖ₚ^p / (‖u‖_q^{αp}‖u‖_r^{p−αp}), a 0-homogeneous quotient whose Euler–Lagrange equation is the bridge: after multiplying u by a suitable scalar, critical points of R_α are exactly the nonzero solutions of the original equation, with μ given by the explicit formula μ_α = α|1−α|^{(p−q)/(r−p)} [R_{α*}(u)]^{(r−q)/(r−p)}. The key parameters are α₀ (the Gagliardo–Nirenberg threshold below which the quotient is degenerate) and α* = q(r−p)/(p(r−q)) (the value where the energy of normalized solutions vanishes). The machinery consists of the Lusternik–Schnirelmann critical levels λ_k(α) and their continuity, monotonicity, and semiconvergence properties.

What would settle it

Compute the Lusternik–Schnirelmann level λ₂(α) on a fixed bounded connected domain for q<r≤p and α∈(0,1): if λ₂(α) equals λ₁(α) or if a sign-changing critical point appears below λ₁(α), the simplicity or sign-classification claims fail. Alternatively, numerically follow the nonnegative solution branch of (1.12) in the convex-concave case: if the branch does not reach a degenerate solution at the μ-threshold, the characterization of degenerate solutions breaks.

Watch

Extended reading notes

Core claim

For bounded Ω and 1 ≤ q < r < p* (p>1), the paper establishes a bijection: after a canonical normalization, every weak solution of −Δₚu = μ|u|^{q−2}u + |u|^{r−2}u (sign on the second term either way) corresponds to a critical point of the homogeneous quotient R_α(u) = ‖∇u‖ₚ^p/(‖u‖_q^{αp}‖u‖_r^{p−αp}), with α = μ‖u‖_q^q/‖∇u‖ₚ^p, and conversely (Lemma 1.1). It then develops the spectral theory of R_α — Lusternik–Schnirelmann critical levels λ_k(α), their continuity and monotonicity in α, and the translation level μ_α — and applies this to characterize degenerate solutions of the convex-concave problem, simplicity and isolation of the ground state in the subhomogeneous case, and the absence of

Load-bearing premise

The classification theorems assume a connected (and for isolation, C^{1,θ}-regular) bounded domain, and the variational construction of eigenvalues requires α > α₀, the Gagliardo–Nirenberg threshold; below α₀ the quotient degenerates and λ₁(α) = 0.

Editorial extensions

If this is right

  • In the convex-concave case q < p < r, all degenerate solutions — those where the second directional derivative of the energy vanishes — are exactly the critical points of R_α with α = (r−p)/(r−q); since λ_k(α) yields infinitely many such critical points, there are infinitely many degenerate solutions.
  • In the subhomogeneous case q < r ≤ p on bounded connected domains, λ₁(α) is simple and isolated for α ∈ [0,1], minimizers are the only nonnegative critical points, and any critical point above λ₁(α) changes sign.
  • In the superhomogeneous case p < q < r, critical points below a small neighborhood of λ₁(α) are sign-constant, so any bifurcation from the ground state must begin with sign-constant ones.
  • The translation level μ_α is lower/upper semicontinuous, giving uniform bounds on the threshold μ* in the convex-concave and superhomogeneous cases; these thresholds are attained for k-th Lusternik–Schnirelmann branches.
  • The index k of λ_k(α) provides a classification of solution branches of the original problem.

Reading between the lines

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

  • The bijection implies that the full bifurcation diagram of the equation is encoded in the single family R_α; a natural numerical project is to plot the branches (α, μ_α) for each k, which would visualize the folds predicted by the superhomogeneous threshold M2.
  • Since the quotient R_α only uses L^q and L^r norms, the same dictionary might extend to equations with more than two pure-power terms by using a product of several norms, provided the homogeneity constraint can be made 0-homogeneous.
  • The paper's Remark 7(iv) leaves open whether the threshold μ* is attained at a minimizer of R_{(r−p)/(r−q)}; if so, the largest nonnegative solution of the convex-concave problem is degenerate — a testable consequence.
  • On disconnected domains, the simplicity and classification results fail (as the authors note); one could recover them by working on each component and then combining, suggesting a version of the theory for multiply connected domains.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper investigates the family of 0-homogeneous Rayleigh quotients R_α(u)=‖∇u‖_p^p/(‖u‖_q^{αp}‖u‖_r^{(1-α)p}) on W_0^{1,p}(Ω). Its central observation, formalized in Lemma 1.1, is that for r≠p, nonzero solutions of -Δ_p u = μ|u|^{q-2}u ± |u|^{r-2}u parameterized by μ are in bijection with suitably normalized critical points of R_α parameterized by α, with α = μ‖u‖_q^q/‖∇u‖_p^p. The paper develops the variational theory of R_α: existence of Lusternik–Schnirelmann critical levels λ_k(α) for α>α_0, continuity and monotonicity in α, estimates and linear independence of critical points, the translation level μ_α, and energy identities. For the subhomogeneous case q<r≤p it proves simplicity and isolation of λ_1(α) and that all higher critical points are sign-changing; for the superhomogeneous case p<q<r it shows that near λ_1(α) critical points are sign-constant. It also characterizes degenerate solutions of the convex–concave problem as critical points of R_α for the specific value α=(r-p)/(r-q).

Significance. If correct, the bijection provides a useful dictionary between two parameterizations of the same p-Laplacian problem, allowing existence, multiplicity, and qualitative properties to be transferred from the quotient R_α to the equation. The paper is largely self-contained and uses standard tools (Palais–Smale condition, Lusternik–Schnirelmann theory, hidden convexity, strong maximum principle) with explicit domain and parameter assumptions. The identification of degenerate solutions with a fixed α is novel and potentially useful. The proofs are detailed and the central derivation (Lemma 1.1) is a direct computation with no fitted parameters or unstated assumptions. The paper also carefully notes limitations, such as the role of connectedness and the α>α_0 restriction for the variational machinery.

minor comments (4)
  1. [Lemma 3.12, proof] The text states 'Recalling that q < p', but the argument only needs q < r. Since the paper's default assumption is q < r, the proof is valid, but the phrase should read 'q < r' to avoid confusion when q>p.
  2. [Lemma 5.1, Eq. (5.4)] The displayed inequality (5.4) is incorrect as written: after raising to q/(αp), the second factor should contain A^{-αr/((1-α)q)} and B^{-αr/((1-α)q)} (with minus signs), and the right-hand side should be 1, not AB. With this correction the AM-GM contradiction immediately follows. The current formula appears to be a typographical error, but it should be fixed before publication.
  3. [Remark 4.12] The assertion that μ* is attained 'in view of Lemma 2.11' is under-justified, since Lemma 2.11 concerns Palais–Smale sequences at a fixed α and does not directly give compactness over the family α∈[0,1). This statement is in a remark and not load-bearing, but it would benefit from a brief justification or a softer phrasing.
  4. [Section 2, notation] The phrase 'properly normalized critical points' is used informally in the introduction and abstract; the precise normalizations (C_α, C'_α, M_α) are introduced later. A forward reference in the abstract or introduction would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central bijection is a direct Euler-Lagrange computation and all load-bearing inputs are external or proved in-line.

full rationale

The central claim (Lemma 1.1) is an algebraic consequence of the Euler-Lagrange equation. Testing (1.12) with u gives (1.14); setting α = μ∥u∥_q^q / ∥∇u∥_p^p exactly transforms (1.12) into (1.3), which is by definition the critical point equation for R_α, and the normalization (1.4) and translation level (1.5) are the same identity. The converse is the same computation in reverse, so the bijection is not an assumption imported from elsewhere. The variational machinery (Lemmas 2.11, 2.12, 3.4, 3.8) is proved in-line from compact embeddings, the Hölder and Gagliardo-Nirenberg inequalities, and standard Lusternik-Schnirelmann theory; no parameter is fitted and no target result is presupposed. The simplicity/isolation results in Section 5 use hidden convexity, Hopf's lemma, and the external uniqueness result [20]; [11] and [12] are cited only for auxiliary antimaximum/dead-core facts that are not the paper's main conclusion and do not encode the bijection. The paper even flags its own assumptions (Remark 5.6 on connectedness) and under-justified remarks (Remark 4.12), which supports the assessment that no result is being forced by definition or by self-citation. A typo in Lemma 3.12 ('q<p' should read 'q<r') affects exposition, not circularity.

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

The paper introduces no free parameters fitted to data: α is a mathematical parameter indexing the quotient, and the special values α0, α* are derived from embedding constants and the equation coefficients. The translation level μ_α is defined from the critical point rather than fitted. Auxiliary results from the literature are used as black boxes; no new axiomatic entities are postulated.

assumptions (6)
  • standard math Sobolev embedding and Rellich-Kondrachov compactness for W_0^{1,p}(Ω) into L^r(Ω), r<p*
    Used in Lemma 2.1 and throughout to pass from weak to strong convergence in Lebesgue norms.
  • standard math Gagliardo-Nirenberg interpolation inequality (1.21) with sharp constants from Del Pino-Dolbeault, Liu-Wang, Weinstein
    The threshold α0 (2.1) is defined via the GN parameter θ; used in Lemma 2.1 and Proposition 2.5 to establish positivity and attainability of λ1(α).
  • standard math Lusternik-Schnirelmann theory and Krasnoselskii genus properties
    Used in Lemma 2.12 to guarantee that λk(α) are critical levels; explicitly deferred to [26, Theorem 5.2].
  • standard math C^{1,ϑ} elliptic regularity (DiBenedetto, Tolksdorf, Lieberman) and Hopf's lemma
    Needed in Remark 2.3 and Lemma 5.7 for strong convergence in C^1 and the isolatedness of λ1(α).
  • standard math Strong maximum principle for the p-Laplacian (Vázquez)
    Used in Lemma 2.8 to conclude that minimizers of λ1(α) are sign-definite.
  • standard math Díaz-Saá hidden convexity inequality and uniqueness of positive solutions for certain sub/superhomogeneous problems ([20, Lemma 2], [11, Prop 2.17])
    The uniqueness results are for the specific PDEs (5.5)/(5.6) with given parameter ranges; used in Lemmas 5.2, 5.5, 6.4 to rule out coincident nonnegative solutions.

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Pith. "Pith review of On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities." pith.science (2026). https://pith.science/paper/6ZDGNYKB

@misc{pith2026251110199,
  author       = {Pith},
  title        = {Pith review of: On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZDGNYKB}},
  note         = {Machine review of arXiv:2511.10199}
}
abstract

Let $\Omega$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\Omega)$-solutions of the equation $-\Delta_p u = \mu |u|^{q-2}u + |u|^{r-2}u$ parameterized by $\mu$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_\alpha(u)=\|\nabla u\|_p^p/ (\|u\|_q^{\alpha p} \|u\|_r^{p-\alpha p})$ parameterized by $\alpha$. We study this bijection and properties of $R_\alpha$ for various relations between $p,q,r$. In particular, for the generalized convex-concave problem (the case $q<p<r$) the bijection allows to provide the existence and characterization of all degenerate solutions corresponding to the inflection point of the fibred energy functional: they are critical points of $R_\alpha$ exclusively with $\alpha = (r-p)/(r-q)$. In the subhomogeneous case $q<r \leq p$ and under additional assumptions on $\Omega$, the ground state level of $R_\alpha$ is simple and isolated, and minimizers of $R_\alpha$ exhaust the whole set of sign-constant solutions of the corresponding equation. In the superhomogeneous case $p < q<r$, there are no sign-changing critical points in a vicinity of the ground state level of $R_\alpha$.

Figures

Figures reproduced from arXiv: 2511.10199 by the authors.

Figure 1
Figure 1. Schematically depicted results of Propositions [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Schematically depicted results of Propositions [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Schematically depicted bifurcation diagrams for [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Schematically depicted behavior of t 7→ Eµα (tu) for some u ∈ W 1,p 0 (Ω) \ {0} and various values of µα. A curve with the critical point of inflection type is distinguished. In the following two lemmas, we show that the value of α characterizes the signs of Eµα (uα) a…
Figure 5
Figure 5. Figure 5: Schematically depicted results of Propositions [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]

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