Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations

Jianwen Zhou; Puming Yang

REVIEW 4 major objections 5 minor 24 references

Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The p-Kirchhoff equation with prescribed mass has a complete existence-nonexistence-multiplicity picture, governed by two critical exponents and a sharp Gagliardo-Nirenberg optimizer.

desk verdict Genuinely first full normalized-solution picture for the p-Kirchhoff equation in R^3, but the paper ships a wrong proof of a load-bearing lemma that is nevertheless repairable. read the letter →

arxiv 2411.11037 v2 pith:VLKY3FF6 submitted 2024-11-17 math.AP

Jianwen Zhou , Puming Yang This is my paper · ORCID

classification math.AP MSC 35J2035J9235B40

keywords p-KirchhoffequationnormalizedsolutionsprescribedL^pmassgroundstatemultiplicityasymptoticbehaviorGagliardo-Nirenberginequalityp-Laplacian

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 studies the p-Kirchhoff equation $-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^p dx\right)\Delta_p u=\lambda|u|^{p-2}u+|u|^{q-2}u$ in $\mathbb{R}^3$ with prescribed $L^p$ norm $\|u\|_p=c$, where $3/20$, whether constrained critical points exist, how many there are, and what happens as the nonlocal coefficient $b$ tends to $0$. The answer depends on where $q$ sits relative to the thresholds $p+p^2/3$ and $p+2p^2/3$: minimizers of the energy exist exactly above stated masses in the subcritical and intermediate regimes, no minimizer exists at $q=p+2p^2/3$, and in the supercritical range there is a radial ground state for every $c>0$ together with infinitely many radial solutions whose energies tend to $+\infty$. The sharp formulas for the thresholds and explicit minimizers are built on a Gagliardo-Nirenberg inequality of p-Laplacian type and its optimizer $Q$; when $3/2

What carries the argument

The load-bearing object is the sharp Gagliardo-Nirenberg inequality of p-Laplacian type (Lemma 2.1), which bounds $\|u\|_q$ by a power of $\|\nabla u\|_p$ times a power of $\|u\|_p$, with the sharp constant expressed through the ground state $Q$ of the auxiliary equation (2.2). This inequality supplies the lower bounds that reduce the constrained energy to a one-variable function of $t=\|\nabla u\|_p^p$; all threshold masses, including $a^{3/p^2}\|Q\|_p$, $c_*$, and the critical-mass condition in Lemma 3.2, are read off from that reduced function. The other central mechanism is the Pohozaev identity $P(u)=a\|\nabla u\|_p^p+b\|\nabla u\|_p^{2p}-\frac{3(q-p)}{pq}\|u\|_q^q=0$, the integral relation any weak solution must satisfy; it defines the natural constraint manifold $\mathcal{M}(c)$ used in the supercritical case and forces $\lambda<0$ for every solution. In the multiplicity proof, a scaling map $k(u,\theta)=e^{3\theta/p}u(e^{\theta}x)$ and a finite-dimensional linking argument produce Palais-Smale sequences with $P(u_k)\to0$; convergence of those sequences is what yields the high-energy radial solutions.

What would settle it

Numerically minimize the constrained energy for one fixed $p\in(3/2,2)$ and $q=p+2p^2/3$ across a range of masses $c$: Theorem 1.1(2) predicts the infimum is $0$ for $c$ below the stated threshold and $i(c)=-\infty$ above, with no minimizer in either case. Finding a finite attained minimum for any $c>0$ would falsify the central nonexistence claim.

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Extended reading notes

Core claim

The central discovery is that the constrained minimization problem for the energy $I(u)=\frac{a}{p}\int_{\mathbb{R}^3}|\nabla u|^p\,dx+\frac{b}{2p}\left(\int_{\mathbb{R}^3}|\nabla u|^p\,dx\right)^2-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx$ on the sphere $\|u\|_p=c$ is governed by a sharp dichotomy in $q$. For $p<q<p+p^2/3$, $i(c)$ has a minimizer for every $c>0$; for $q=p+p^2/3$, it has a minimizer exactly for $c>a^{3/p^2}\|Q\|_p$; for $p+p^2/3<q<p+2p^2/3$, there is a critical mass $c_*>0$ such that minimizers exist exactly for $c\ge c_*$; and for $q=p+2p^2/3$, no minimizer exists for any $c>0$. In the supercritical range $p+2p^2/3<q<p^*$, where $I$ is unbounded below on $\mathcal{S}(c)$, the paper works on the radial space and obtains, for every $c>0$, a radial ground state and a sequence of radial solutions with $\|u_n\|_{W^{1,p}_r}\to+\infty$ and $I(u_n)\to+\infty$; as $b\to0^+$, these solutions converge in $W^{1,p}_r(\mathbb{R}^3)$ to weak solutions of the limiting p-Laplacian equation $-a\Delta_p u-\lambda|u|^{p-2}u=|u|^{q-2}u$. For $3/2<p\le2$, the unique minimizer is explicit: $u_c=c\,\mu_q^{3/p}\|Q\|_p^{-1}Q(\mu_q x)$ with $\mu_q$ determined by a one-variable function $f_q$, and the mountain-pass value $\gamma(c)=f_q(t_q)$ is attained by the same scaling.

Load-bearing premise

The paper's thresholds, explicit minimizers, and uniqueness claims all rest on a sharp interpolation inequality whose optimizer is the ground state $Q$; for $3/2<p\le2$, it also relies on $Q$ being unique up to translation, and these external facts are cited rather than reproved.

Editorial extensions

If this is right

For $p<q<p+p^2/3$, every prescribed mass $c>0$ admits a minimizer, so the nonlocal Kirchhoff term does not destroy the subcritical existence picture.
At $q=p+p^2/3$ and in $p+p^2/3<q<p+2p^2/3$, minimizers exist only above explicit mass thresholds, namely $c>a^{3/p^2}\|Q\|_p$ and $c\ge c_*$ respectively.
At $q=p+2p^2/3$, the infimum is never attained for any $c>0$; any normalized solution at this critical exponent must come from a different variational principle such as the mountain-pass level of Theorem 1.8.
In the supercritical range $p+2p^2/3<q<p^*$, radial ground states exist for every $c>0$, and there are infinitely many radial solutions with energies tending to $+\infty$.
As $b\to0^+$, each of these radial solutions converges (up to subsequence) to a weak solution of the limiting p-Laplacian equation with the same prescribed mass.

Reading between the lines

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

An implication the paper leaves implicit is that, for $3/2<p\le2$, the explicit minimizer formula forces every subcritical and minimizer solution to be a scaling of the single optimizer $Q$; uniqueness there is structural rather than accidental.
A testable extension is to measure the rate of convergence in $b\to0^+$ for the first few radial solutions; the proof gives $W^{1,p}_r$ convergence but no rate, and the rate may depend on the energy level.
The critical exponent $q=p+2p^2/3$ is exactly where the scaling exponent $3(q-p)/p$ in the Pohozaev term crosses the exponent $2p$ of the nonlocal term; the same crossing should produce analogous existence/nonexistence dichotomies in other nonlocal problems.
For $p>2$, the missing uniqueness of the optimizer $Q$ blocks the explicit formulas; if that uniqueness were established, the same argument would extend Theorems 1.6 and 1.8 beyond $p\le2$.

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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

Two things to know about arXiv:2411.11037. First, it is genuinely the first paper to handle the p-Kirchhoff equation with prescribed L^p mass across the full subcritical-critical-supercritical range, and it gives a fairly complete picture: sharp mass thresholds for minimizers, nonexistence at the L^p-critical exponent, radial ground states and a high-energy sequence in the supercritical case, and a b→0+ limit to the p-Laplacian. Second, the proof of Lemma 3.4, which rules out dichotomy and is load-bearing for Theorem 1.1, is printed incorrectly: the scaling identity in (3.9) does not produce the claimed inequality, and the final display is tautological. The lemma is still true and easily repairable — applying the dilation v = θu gives i(θc) < θ^p i(c) when i(c) < 0, hence c ↦ i(c)/c^p is strictly decreasing, and subadditivity follows. So the paper's main argument survives, but the written proof needs rewriting.

What is good: the variational scheme is standard and executed with care. The use of the sharp GN inequality with optimizer Q is honest; the thresholds, explicit minimizers, and mountain-pass levels are all expressed through Q, not through quantities of the target equation, so there is no circularity. The uniqueness statements for 3/2 < p ≤ 2 properly rely on the known uniqueness of Q, and the paper acknowledges the p > 2 obstruction. The b→0+ asymptotic section is a nice add-on, and the multiplicity argument follows the Jeanjean-Bartsch linking framework cleanly.

Soft spots: aside from Lemma 3.4, there is a typo in the proof of Theorem 1.1(3) where a sequence is claimed to lie in S(c) but should be in S(c_n). Several auxilary lemmas are omitted by reference; that is acceptable if the cited analogues really match, but the authors should spell out the adaptation for the nonlocal term. The dependence on the sharp GN inequality is an external input, and if equality cases for p > 2 were not unique, the explicit formulas in Theorem 1.6 would not extend; the paper is clear about that.

Bottom line: this is a serious paper for the normalized-solutions community. It deserves a real referee and will probably be cited once the proof of Lemma 3.4 is corrected. I would send it to peer review, but require the authors to replace that proof and fix the small errors before it is accepted.

Referee Report

4 major / 5 minor

Summary. The paper studies the p-Kirchhoff equation with prescribed L^p mass in R^3, for 3/2<p<3 and p<q<p*. It claims: (i) sharp existence/nonexistence of constrained minimizers of the energy in the L^p-subcritical and L^p-critical ranges, with no minimizer at the L^p-critical exponent; (ii) existence of a radial ground state and of infinitely many radial solutions with diverging energy in the L^p-supercritical range; (iii) convergence of those solutions, as the Kirchhoff coefficient b tends to 0+, to solutions of a p-Laplacian equation; and (iv) uniqueness and explicit formulas for minimizers when 3/2<p<=2. The proofs combine the sharp Gagliardo-Nirenberg inequality of p-Laplacian type, minimization on S(c) and on a Pohozaev manifold, Ekeland's principle, and a Jeanjean-type linking construction.

Significance. If the results hold, the paper gives a fairly complete existence-nonexistence-multiplicity picture for normalized solutions of this p-Kirchhoff problem, extending known p=2 results to the quasilinear setting. The explicit thresholds and minimizers are expressed through the optimizer Q of the Gagliardo-Nirenberg inequality, and the b->0+ asymptotic statement is a useful addition. The variational scheme is standard and the main theorems are plausible. However, several load-bearing proofs contain gaps that need to be repaired: the strict subadditivity in Lemma 3.4 is not proved as written, the displayed critical-mass formula c_* contains an algebraic error, and the radial compactness argument in Theorem 1.3 is not justified. These issues are local and appear repairable, but they affect the central existence and threshold claims.

major comments (4)

[Lemma 3.4] The proof of the strict subadditivity is invalid as printed. The final display reads i(c) = (alpha^p/c^p) i(c/alpha * alpha) + ((c^p-alpha^p)/c^p) i(beta * c/beta) < i(alpha)+i(beta), but c/alpha*alpha = c and c/beta*beta = c, so the equality is the tautology i(c)=i(c). Applying (3.9) with theta=c/alpha and theta=c/beta gives two inequalities of the form i(c)<(c/alpha)^p i(alpha) and i(c)<(c/beta)^p i(beta); adding them gives 2i(c)<i(alpha)+i(beta), not i(c)<i(alpha)+i(beta). Since (3.19) in the proof of Theorem 1.1 uses exactly this strict subadditivity to rule out dichotomy, the printed argument is incomplete. A repair is available: (3.9) implies that c -> i(c)/c^p is strictly decreasing on the range satisfying i(c)<0; applying this monotonicity to alpha and beta and adding the weighted inequalities yields i(c)<i(alpha)+i(beta). The lemma should be rewritten accordingly.
[Lemma 3.1(4) and Theorem 1.1(iii)] The displayed formula for c_* contains an algebraic error. With p_2 = 3(q-p)/p^2 - 1, one has 2p p_2 = (6q - 6p - 2p^2)/p, so the factor coming from the Young inequality is (bp/(6q - 6p - 2p^2))^{p_2}, not (bp/(6pq - 8p^2))^{p_2}. For p=2, q=4 these two expressions differ by a factor of 4. Since c_* is the sharp threshold in Theorem 1.1(iii) and in Lemma 3.1, this is not a harmless typo: as printed, the threshold formula is wrong. Please correct the denominator in Theorem 1.1, Lemma 3.1, and anywhere else the formula appears.
[Proof of Theorem 1.3] After finding y_n such that the integral over B_1(y_n) is positive, the proof sets u_n(x)=v_n(x+y_n) and states that {u_n} is a minimizing sequence for m(c). This is false when y_n is nonzero: translations of radial functions are not radial, while M(c) is contained in S_r(c). The subsequent weak limit is claimed in W^{1,p}_r(R^3), which requires the sequence to be radial. If the y_n are unbounded, the translated sequence may even converge weakly to zero. The proof should instead use the Strauss radial compactness theorem to obtain strong L^q convergence directly from the radial minimizing sequence, or otherwise justify the use of translations without destroying radial symmetry.
[Lemma 5.8(ii)] The inference 'Since |u_k|_q^q -> |u|_q^q != 0, we obtain lambda < 0' is not justified. Equation (5.11) gives only lambda <= 0. If lambda = 0, then the limiting equation (5.14) combined with the Nehari identity (5.16) and the Pohozaev identity would force q = p*, contradicting the assumption q < p*. This missing argument is needed both for the negativity of the Lagrange multiplier and for the strong convergence step (5.17). Please add the argument or cite the analogous reasoning used later in Theorem 1.5.

minor comments (5)

[Throughout] There are several typographical errors, including 'Gargliardo' in the abstract, 'rencent' in Section 1, 'exsists' in the proof of Theorem 1.1, and 'Fisrtly' in Section 5; these should be corrected.
[Theorem 1.1 and Lemma 3.1] The displayed formula for c_* has an unmatched parenthesis after 2p^2 - 3q + 3p; please typeset the formula with consistent parentheses and exponents.
[Theorem 1.6] In the proof of Theorem 1.6, the coefficient alpha is printed as c/|Q|_P; the subscript should be p.
[Lemma 3.1(4)] The sentence defining t_0 after the Young inequality is garbled in the typesetting; please rewrite it so that the equality condition in (3.7) is stated clearly.
[Lemma 3.4] The notation for the p-th root appears as 'p√' in the statement of Lemma 3.4; use a proper root symbol such as \sqrt[p]{\cdot}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: load-bearing inputs are the external sharp Gagliardo–Nirenberg inequality and standard variational machinery; no conclusion is fitted to itself or to the authors' own prior work.

full rationale

The derivation chain is not circular. Theorem 1.1 and the subsequent uniqueness, multiplicity, and asymptotic results are built on the sharp Gagliardo–Nirenberg inequality of p-Laplacian type (Lemma 2.1), cited to Agueh [10] and Weinstein [11], and on the uniqueness of its optimizer for 1 < p ≤ 2, cited to Serrin–Tang [24]. These results are external to the paper: Q is defined as a ground state of equation (2.2), not as a solution of the target p-Kirchhoff equation (1.1), and the thresholds and explicit formulas in Theorems 1.1, 1.6, and 1.8 are computed from Q and from one-variable functions f_q(t) obtained by evaluating the energy on scalings of Q. The upper and lower bounds for i(c) and γ(c) are independent: the lower bounds use the external inequality, while the upper bounds use explicit test functions, so no quantity is fitted to the statement being proved. The paper contains no fitted parameters, no load-bearing self-citations, and no renamed empirical pattern. One non-circular rigor concern should be noted: the printed proof of Lemma 3.4 ends with a tautological display, i(c) = (alpha^p/c^p)i(c) + ((c^p-alpha^p)/c^p)i(c), and the claimed strict subadditivity does not follow from the preceding scaling inequality (3.9) as written; the lemma is repairable by working with the monotonicity of i(c)/c^p. This is a correctness gap in the compactness argument for Theorem 1.1, but it is not circularity, because the lemma's conclusion is not built into its hypotheses.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rest on four imported analytical pillars: the sharp GN inequality with equality cases, radial compactness, the radial Schauder-basis spectral-gap lemma, and the Pohozaev identity. None of these is invented by the present authors, and no new entities (forces, particles, dimensions) are introduced.

assumptions (5)

domain assumption Sharp Gagliardo-Nirenberg inequality of p-Laplacian type (Lemma 2.1) with explicit constant and, for 1<p<=2, uniqueness of the optimizer Q up to translations.
This is the main external input; it defines |Q|_p and the thresholds in Theorems 1.1, 1.6, 1.8. Cited to [10], [11], [24].
standard math Radial compactness: the embedding W_r^{1,p}(R^3) into L^q(R^3) is compact for p<q<p*.
Used in the supercritical sections (Theorems 1.3, 1.4, 1.5) to recover compactness in the radial setting; standard Strauss-type result.
domain assumption Schauder basis of W_r^{1,p}(R^3) with the spectral-gap property mu_n -> +infinity (Lemma 5.1).
Imported from Zhang-Zhang [19], Lemma 6.1; this is the spectral-gap input for the linking and multiplicity argument.
standard math Pohozaev identity for solutions of (1.1)-(1.2) and the conclusion that the Lagrange multiplier satisfies lambda<0 (Lemma 2.2).
The proof is omitted and referenced to [12] and [6]; it is used throughout to restrict solutions to the Pohozaev manifold and to constrain lambda.
standard math Ekeland's variational principle and standard minimax theory on the C^1 manifold S(c).
Used in Lemma 3.5 and Lemma 5.7; standard functional analysis.

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Pith. "Pith review of Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations." pith.science (2026). https://pith.science/paper/VLKY3FF6

@misc{pith2026241111037,
  author       = {Pith},
  title        = {Pith review of: Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLKY3FF6}},
  note         = {Machine review of arXiv:2411.11037}
}

abstract

In this paper, we study a type of p-Kirchhoff equation $$ -\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \varDelta _pu=\lambda \left| u \right|^{p-2}u+\left| u \right|^{q-2}u, x \in \mathbb{R}^3 $$ with the prescribed mass $$ \left(\int_{\mathbb{R} ^3}{\left| u \right|^{p}dx}\right)^\frac{1}{p} = c > 0 $$ where $a>0, b > 0,\frac{3}{2} <p <3, p < q < p^{\ast}:=\frac{3p}{3-p} $,$\varDelta _pu=div\left( \left| \nabla u \right|^{p-2}\nabla u \right)$ is the $p$-Laplacian of $u$, $\lambda \in \mathbb{R}$ is Lagrange multiplier. We consider both $L^p$-subcritical , $L^p$-critical and $L^p$-supercritical cases. Precisely, in the $L^p$-subcritical and $L^p$-critical cases, we obtain the existence and nonexistence of the normalized solutions for the $p$-Kirchhoff equation. In the $L^p$-supercritical case, we obtain the existence of radial ground sates and multiplicity of radial normalized solutions for the $p$-Kirchhoff equation. Furthermore, we study the asymptotic behavior of normalized solutions when $b \rightarrow 0^+$. Besides, when $\frac{3}{2} < p \leq 2$, benefit from the uniqueness(up to translations) of optimizer for Gargliardo-Nirenberg inequality, we show the existence and uniqueness of normalized solutions and provide the accurate descriptions.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

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P. Pucci, M. Xiang, and B. Zhang, “Existence and multiplicity of entire solutions for fractional p-kirchhoff equations,” Advances in Nonlinear Analysis , vol. 5, no. 1, pp. 27–55, 2016

work page 2016

[2] [2]

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F. J. S. Corrˆ ea and G. M. Figueiredo, “On a p-kirchhoff equation via krasnoselskii’s genus,”Applied Mathematics Letters, vol. 22, no. 6, pp. 819–822, 2009

work page 2009

[3] [3]

Multiple solutions for a kirchhoff equation with critical growth,

M. F. Furtado, L. D. de Oliveira, and J. P. P. da Silva, “Multiple solutions for a kirchhoff equation with critical growth,” Zeitschrift f¨ ur angewandte Mathematik und Physik, vol. 70, pp. 1–15, 2019

work page 2019

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Kirchhoff, Vorlesungen ¨ uber mathematische Physik, vol

G. Kirchhoff, Vorlesungen ¨ uber mathematische Physik, vol. 1. 1883

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J.-L. Lions, “On some questions in boundary value problems of mathematical physics,” in North- Holland Mathematics Studies , vol. 30, pp. 284–346, Elsevier, 1978

work page 1978

[6] [6]

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H. Ye, “The sharp existence of constrained minimizers for a class of nonlinear kirchhoff equations,” Mathematical Methods in the Applied Sciences , vol. 38, no. 13, pp. 2663–2679, 2015

work page 2015

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X. Zeng and Y. Zhang, “Existence and uniqueness of normalized solutions for the kirchhoff equa- tion,” Applied Mathematics Letters , vol. 74, pp. 52–59, 2017

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Q. Lou and Z. Zhang, “Multiplicity and concentration of normalized solutions to p-laplacian equa- tions,” Zeitschrift f¨ ur angewandte Mathematik und Physik, vol. 75, no. 3, p. 81, 2024

work page 2024