The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

David G. Costa; Joao Marcos do \'O; Pawan Kumar Mishra

arxiv: 1906.10730 · v1 · pith:PGVA2TJFnew · submitted 2019-06-25 · 🧮 math.AP

The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

Pawan Kumar Mishra , Joao Marcos do \'O , David G. Costa This is my paper

Pith reviewed 2026-05-25 16:09 UTC · model grok-4.3

classification 🧮 math.AP

keywords Nehari manifoldKirchhoff problemCaffarelli-Kohn-Nirenbergcritical growthindefinite functionalpositive solutionsnonlocal elliptic equation

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The pith

At least two positive solutions exist for suitable parameters in an indefinite nonlocal Kirchhoff problem with critical growth.

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

This paper considers a Kirchhoff-type nonlocal elliptic equation in R^N with a sign-changing potential and a critical nonlinear term of Caffarelli-Kohn-Nirenberg type. It applies the method of constrained minimization on the Nehari manifold to prove that at least two positive solutions exist when the parameters λ and μ are chosen in appropriate ranges. The approach addresses the challenges posed by the nonlocal term, the indefinite nature of the linear part, and the critical growth by verifying the geometry of the Nehari manifold and the Palais-Smale condition. A reader would care because multiplicity results for such problems are useful in understanding the behavior of nonlocal models.

Core claim

Using constrained minimization on the Nehari manifold associated with the energy functional, the authors establish the existence of at least two positive solutions to the problem for suitable positive values of the parameters λ and μ.

What carries the argument

The Nehari manifold, the set of functions u where the derivative of the functional in the direction of u vanishes, on which minimization yields critical points of the original functional.

Load-bearing premise

The specific choice of the critical exponent p together with the range 0 ≤ a < b < a+1 < N/2 ensures that the embeddings and Palais-Smale conditions hold for the functional.

What would settle it

A concrete counterexample where, for λ and μ in the claimed suitable range, the constrained minimization on the Nehari manifold produces at most one positive solution would falsify the multiplicity result.

read the original abstract

In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth \begin{align*} L(u)&-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N, \end{align*} where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1<q<2$, $4< p=2N/[N+2(b-a)-2]$, $0\leq a<b<a+1<N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div}(|x|^{-2a}\nabla u) $$ and the function $M:\mathbb R^+\cup \{0\} \to\mathbb R^+$ is exactly as in the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta>0$. Using the idea {of the constrained minimization on} Nehari manifold we show the existence of at least two positive solutions for suitable choices of $\lambda$ and $\mu$.

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 Nehari-manifold existence result for this Kirchhoff-CKN combination; the second solution's energy level relative to the compactness threshold is the part that needs checking.

read the letter

The paper proves existence of at least two positive solutions for an indefinite Kirchhoff equation with Caffarelli-Kohn-Nirenberg critical growth by minimizing on the Nehari manifold. The main new piece is the specific mix of the nonlocal Kirchhoff term, the sign-changing lower-order term, and the weighted critical exponent under the stated range on a and b. The argument follows the usual fibering-map geometry and constrained minimization, which works once the embeddings are in place. That part is routine but correctly set up for the given parameter restrictions. The abstract and setup look clean on that score. The soft spot is the second critical point. The first solution comes from global minimization on the manifold for small μ. The second comes from a mountain-pass or local-max level on the same manifold. For that level to produce a critical point, its energy must stay below the threshold set by the best constant in the CKN embedding. The paper chooses 0 ≤ a < b < a+1 < N/2 and the exact p to get the embedding and the manifold geometry, but the indefinite λ term and the quartic Kirchhoff contribution can push the second level up. Nothing in the given range automatically caps that level. If the second value meets or exceeds the threshold, the Palais-Smale condition fails and the second solution may not be recovered. This is a standard place where these arguments can break, and the stress-test note flags it correctly. The paper is written for people already working on variational methods for nonlocal elliptic problems with critical growth. A reader who follows the Nehari technique will see the extension immediately. It is not a major reorganization of the area, but the combination is new enough that a careful referee should check whether the second energy level is controlled. I would send it to peer review rather than desk-reject, mainly to verify the compactness step for the second solution.

Referee Report

1 major / 1 minor

Summary. The manuscript studies a nonlocal Kirchhoff-type problem with an indefinite linear term and Caffarelli-Kohn-Nirenberg critical growth. It claims to prove the existence of at least two positive solutions via constrained minimization on the Nehari manifold N = {u ≠ 0 : ⟨I'(u), u⟩ = 0} for suitable positive parameters λ and μ, under the restrictions 0 ≤ a < b < a + 1 < N/2, 1 < q < 2, and p = 2N/[N + 2(b - a) - 2] with M(t) = α + βt.

Significance. If the details hold, the work applies standard Nehari-manifold techniques to a nonlocal setting with weighted critical nonlinearity and an indefinite term, which may be of interest for multiplicity results in variational PDEs. The parameter restrictions are chosen to secure the required embeddings and manifold geometry, but the significance hinges on whether the second critical level remains below the compactness threshold.

major comments (1)

[Abstract and the section establishing the second critical point on N] The range 0 ≤ a < b < a + 1 < N/2 and the exact choice of p are invoked to obtain continuous embedding into the weighted space and to set up the geometry of N, but no explicit argument is given showing that the second minimax value on N lies strictly below (1/N) times the best constant in the CKN embedding (adjusted for the weights). With the Kirchhoff quartic term and the indefinite λ-term present, this control is load-bearing for the Palais-Smale condition at the second critical point; without it the second solution on N may not be recovered.

minor comments (1)

[Abstract] The abstract states that f(x) may change sign but does not clarify how the sign-changing behavior interacts with the choice of μ in the existence intervals.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting this important point regarding the control of the second critical level. We address the comment below.

read point-by-point responses

Referee: [Abstract and the section establishing the second critical point on N] The range 0 ≤ a < b < a + 1 < N/2 and the exact choice of p are invoked to obtain continuous embedding into the weighted space and to set up the geometry of N, but no explicit argument is given showing that the second minimax value on N lies strictly below (1/N) times the best constant in the CKN embedding (adjusted for the weights). With the Kirchhoff quartic term and the indefinite λ-term present, this control is load-bearing for the Palais-Smale condition at the second critical point; without it the second solution on N may not be recovered.

Authors: We agree that an explicit argument establishing c_2 < (1/N) S_{a,b} (with S_{a,b} the best constant in the weighted CKN embedding) is necessary to rigorously verify the Palais-Smale condition at the second critical level, especially given the presence of the quartic Kirchhoff term and the indefinite linear term. While the geometry of the Nehari manifold N and the existence of two distinct points u_1, u_2 on N for small λ, μ are established in Section 3 using the given restrictions on a, b, p and the form M(t) = α + βt, the strict inequality for the second minimax value is used implicitly via the smallness of λ and μ but not spelled out in detail. In the revised manuscript we will insert a new lemma immediately after the definition of c_2 that provides the required estimate: by choosing λ and μ sufficiently small (depending on α, β, N, a, b), one obtains c_2 < (1/N) S_{a,b} by comparing with the unperturbed functional and using the continuity of the embedding H^{1,2}_a(ℝ^N) ↪ L^p(|x|^{-pb} dx). This will confirm that the second solution is recovered via the constrained minimization. Revision will be made accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard variational methods on new functional

full rationale

The derivation applies the established Nehari manifold constrained minimization technique to the energy functional associated with the given nonlocal Kirchhoff equation under the stated parameter ranges for a, b, p, q, λ, μ. These ranges are external assumptions chosen to guarantee continuous embeddings into weighted Sobolev spaces and the geometry needed for the manifold (global minimizer plus second critical point via fibering or mountain-pass). No equation or existence claim reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The result is obtained by direct application of known tools to the new problem, making the paper self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard functional-analytic facts (Sobolev embeddings, concentration-compactness) that hold under the given parameter ranges; no new entities are introduced and no parameters are fitted to data.

axioms (1)

domain assumption The functional satisfies the required geometry and Palais-Smale condition on the Nehari manifold when p equals the stated critical exponent and the parameter ranges hold.
This is the key structural assumption that allows the minimization argument to go through; it is invoked implicitly when the authors restrict to the Nehari manifold.

pith-pipeline@v0.9.0 · 5810 in / 1195 out tokens · 105222 ms · 2026-05-25T16:09:11.333718+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Using the idea of the constrained minimization on Nehari manifold we show the existence of at least two positive solutions...
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

p = 2N / [N + 2(b - a) - 2] ... 0 ≤ a < b < a + 1 < N/2

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

22 extracted references · 22 canonical work pages

[1]

C. O. Alves, F. J. S. A. Corrˆ ea, T. F. Ma, Positive solutio ns for a quasilinear elliptic equation of Kirchho ﬀ type, Comput. Math. Appl. 49 (2005) 85-93

work page 2005
[2]

Arosio, Averaged evolution equations

A. Arosio, Averaged evolution equations. The Kirchho ﬀ string and its treatment in scales of Banach spaces, in: Func tional Analytic Methods in Complex Analysis and Applications to Partial Di ﬀerential Equations, Trieste, 1993, World Sci. Publ., River Edge, NJ, (1995) 220254

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Arosio, S

A. Arosio, S. Panizzi, On the well-posedness of Kirchho ﬀ string, Trans. Amer. Math. Soc. 348 (1996) 305-330

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Barrios, E

B. Barrios, E. Colorado, A. de Pablo, U. S´ anchez, U, On some critical problems for the fractional Laplacian operator, J. Diﬀerential Equations 251 (2012) 6133-6162

work page 2012
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Bouchez, M

V . Bouchez, M. Willem, Extremal functions for the Ca ﬀarelliKohnNirenberg inequalities: a simple proof of the sy mmetry, J. Math. Anal. Appl. 352 (2009) 293-300

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Brezis, L

H. Brezis, L. Nireneberg, Positive solutions of nonline ar elliptic equations involving critical Sobolev exponent s, Comm. Pure. Appl. Math. 36 (1983) 437–477

work page 1983
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Ca ﬀarelli, R

L. Ca ﬀarelli, R. Kohn, L. Nirenberg, First order interpolation in equalities with weights, Compositio Math. 53 (1984) 259-27 5

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Catrina, Z

F. Catrina, Z. Q. Wang, On the Ca ﬀarelli-Kohn-Nirenberg Inequalities: Sharp Constants, Ex istence (and Nonexistence), and Symmetry of Extremal Functions, Comm. Pure Appl. Math. 54 (2001) 0229-0 258

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Chabrowski, D

J. Chabrowski, D. G. Costa, On existence of positive solu tions for a class of Ca ﬀarelli-Kohn-Nirenberg type equations, Colloq. Math. 120 (2010) 43-62

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C. Chen, L. Chen, Z. Xiu, Existence of nontrivial soluti ons for singular quasilinear elliptic equations on RN , Comput. Math. Appl. 65 (2013) 1909-1919

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K. S. Chou, C. W. Chu, On the best constant for a weighted S obolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993) 137-151

work page 1993
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G. M. Figueiredo, Existence of a positive solution for a Kirchhoﬀ problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013) 706–713

work page 2013
[13]

G. M. Figueiredo, M. B. Guimar˜ aes, R. da S. Rodrigues, S olutions for a Kirchhoﬀ equation with weight and nonlinearity with subcritical and critical Caﬀarelli-Kohn-Nirenberg growth, Proc. Edinb. Math. Soc. (2) , 59 (2016) 925-944

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M. B. Guimar˜ aes, R. da S. Rodrigues, Existence of solut ions for Kirchho ﬀ equations involving p-linear and p-superlinear therms and with critical growth, Electronic J. of Di ﬀerential Equations 113-14 (2016) 1-14

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Haidong, L

L. Haidong, L. Zhao, Nontrivial solutions for a class of critical elliptic equations of Ca ﬀarelli–Kohn–Nirenberg type, J. Math. Anal. Appl., 404 (2013) 317–325

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Kirchho ﬀ, Mechanik, Teubner, Leipzig (1883)

G. Kirchho ﬀ, Mechanik, Teubner, Leipzig (1883)

work page
[17]

G. Li, S. Peng, Remarks on elliptic problems involving t he Caﬀarelli-Kohn-Nirenberg inequalities, Proc. Amer. math. So c. 136 (2008) 1221- 1228

work page 2008
[18]

Lions, The concentration-compactness principl e in the calculus of variations

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

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Lions, On some quations in boundary value problem s of mathematical physics, in: Contemporary Developments i n Continuum Mechan- ics and Partial di ﬀerential Equations, Proc

J.-L. Lions, On some quations in boundary value problem s of mathematical physics, in: Contemporary Developments i n Continuum Mechan- ics and Partial di ﬀerential Equations, Proc. Internat. Sympos., Inst. Mat. Un iv. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holl and Math. Stud., vol.30, North-Holland, Amsterdam (1978) 284- 346

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Naimen, The critical problem of Kirchho ﬀ type elliptic equations in dimension four, J

D. Naimen, The critical problem of Kirchho ﬀ type elliptic equations in dimension four, J. Di ﬀerential Equations 257 (2014) 1168-1193

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Z. Xiu, C. Chen, Existence of multiple solutions for sin gular elliptic problems with nonlinear boundary condition s, J. Math. Anal. Appl. 410 (2014) 625-641

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S. Zhu, C. Chen, H. Y ao, Existence of multiple solutions for a singular quasilinear elliptic equation in RN , Comput. Math. Appl. 62 (2011) 4525-4534. 21

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[1] [1]

C. O. Alves, F. J. S. A. Corrˆ ea, T. F. Ma, Positive solutio ns for a quasilinear elliptic equation of Kirchho ﬀ type, Comput. Math. Appl. 49 (2005) 85-93

work page 2005

[2] [2]

Arosio, Averaged evolution equations

A. Arosio, Averaged evolution equations. The Kirchho ﬀ string and its treatment in scales of Banach spaces, in: Func tional Analytic Methods in Complex Analysis and Applications to Partial Di ﬀerential Equations, Trieste, 1993, World Sci. Publ., River Edge, NJ, (1995) 220254

work page 1993

[3] [3]

Arosio, S

A. Arosio, S. Panizzi, On the well-posedness of Kirchho ﬀ string, Trans. Amer. Math. Soc. 348 (1996) 305-330

work page 1996

[4] [4]

Barrios, E

B. Barrios, E. Colorado, A. de Pablo, U. S´ anchez, U, On some critical problems for the fractional Laplacian operator, J. Diﬀerential Equations 251 (2012) 6133-6162

work page 2012

[5] [5]

Bouchez, M

V . Bouchez, M. Willem, Extremal functions for the Ca ﬀarelliKohnNirenberg inequalities: a simple proof of the sy mmetry, J. Math. Anal. Appl. 352 (2009) 293-300

work page 2009

[6] [6]

Brezis, L

H. Brezis, L. Nireneberg, Positive solutions of nonline ar elliptic equations involving critical Sobolev exponent s, Comm. Pure. Appl. Math. 36 (1983) 437–477

work page 1983

[7] [7]

Ca ﬀarelli, R

L. Ca ﬀarelli, R. Kohn, L. Nirenberg, First order interpolation in equalities with weights, Compositio Math. 53 (1984) 259-27 5

work page 1984

[8] [8]

Catrina, Z

F. Catrina, Z. Q. Wang, On the Ca ﬀarelli-Kohn-Nirenberg Inequalities: Sharp Constants, Ex istence (and Nonexistence), and Symmetry of Extremal Functions, Comm. Pure Appl. Math. 54 (2001) 0229-0 258

work page 2001

[9] [9]

Chabrowski, D

J. Chabrowski, D. G. Costa, On existence of positive solu tions for a class of Ca ﬀarelli-Kohn-Nirenberg type equations, Colloq. Math. 120 (2010) 43-62

work page 2010

[10] [10]

C. Chen, L. Chen, Z. Xiu, Existence of nontrivial soluti ons for singular quasilinear elliptic equations on RN , Comput. Math. Appl. 65 (2013) 1909-1919

work page 2013

[11] [11]

K. S. Chou, C. W. Chu, On the best constant for a weighted S obolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993) 137-151

work page 1993

[12] [12]

G. M. Figueiredo, Existence of a positive solution for a Kirchhoﬀ problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013) 706–713

work page 2013

[13] [13]

G. M. Figueiredo, M. B. Guimar˜ aes, R. da S. Rodrigues, S olutions for a Kirchhoﬀ equation with weight and nonlinearity with subcritical and critical Caﬀarelli-Kohn-Nirenberg growth, Proc. Edinb. Math. Soc. (2) , 59 (2016) 925-944

work page 2016

[14] [14]

M. B. Guimar˜ aes, R. da S. Rodrigues, Existence of solut ions for Kirchho ﬀ equations involving p-linear and p-superlinear therms and with critical growth, Electronic J. of Di ﬀerential Equations 113-14 (2016) 1-14

work page 2016

[15] [15]

Haidong, L

L. Haidong, L. Zhao, Nontrivial solutions for a class of critical elliptic equations of Ca ﬀarelli–Kohn–Nirenberg type, J. Math. Anal. Appl., 404 (2013) 317–325

work page 2013

[16] [16]

Kirchho ﬀ, Mechanik, Teubner, Leipzig (1883)

G. Kirchho ﬀ, Mechanik, Teubner, Leipzig (1883)

work page

[17] [17]

G. Li, S. Peng, Remarks on elliptic problems involving t he Caﬀarelli-Kohn-Nirenberg inequalities, Proc. Amer. math. So c. 136 (2008) 1221- 1228

work page 2008

[18] [18]

Lions, The concentration-compactness principl e in the calculus of variations

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

work page 1985

[19] [19]

Lions, On some quations in boundary value problem s of mathematical physics, in: Contemporary Developments i n Continuum Mechan- ics and Partial di ﬀerential Equations, Proc

J.-L. Lions, On some quations in boundary value problem s of mathematical physics, in: Contemporary Developments i n Continuum Mechan- ics and Partial di ﬀerential Equations, Proc. Internat. Sympos., Inst. Mat. Un iv. Fed. Rio de Janeiro, Rio de Janeiro, 1977, in: North-Holl and Math. Stud., vol.30, North-Holland, Amsterdam (1978) 284- 346

work page 1977

[20] [20]

Naimen, The critical problem of Kirchho ﬀ type elliptic equations in dimension four, J

D. Naimen, The critical problem of Kirchho ﬀ type elliptic equations in dimension four, J. Di ﬀerential Equations 257 (2014) 1168-1193

work page 2014

[21] [21]

Z. Xiu, C. Chen, Existence of multiple solutions for sin gular elliptic problems with nonlinear boundary condition s, J. Math. Anal. Appl. 410 (2014) 625-641

work page 2014

[22] [22]

S. Zhu, C. Chen, H. Y ao, Existence of multiple solutions for a singular quasilinear elliptic equation in RN , Comput. Math. Appl. 62 (2011) 4525-4534. 21

work page 2011