Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$

Bo\v{s}tjan Gabrov\v{s}ek; Jijiang Sun; Lin Li; Matija Cencelj

arxiv: 1907.01888 · v1 · pith:3ZDOFQHNnew · submitted 2019-07-03 · 🧮 math.AP

Infinitely many sign-changing solutions for Kirchhoff type problems in mathbb{R}³

Jijiang Sun , Lin Li , Matija Cencelj , Bo\v{s}tjan Gabrov\v{s}ek This is my paper

Pith reviewed 2026-05-25 10:05 UTC · model grok-4.3

classification 🧮 math.AP

keywords Kirchhoff type problemsign-changing solutionsinfinitely many solutionsinvariant setsLjusternik-Schnirelmann methodsuperlinear nonlinearitysubcritical growthR^3

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The pith

When the nonlinearity is odd the Kirchhoff problem in R^3 has infinitely many sign-changing solutions

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

This paper examines the Kirchhoff-type equation with a continuous coercive potential on all of R^3. It establishes that an odd nonlinearity that is superlinear at infinity and subcritical produces infinitely many sign-changing solutions. The argument combines the invariant sets method with a Ljusternik-Schnirelmann minimax procedure on the energy functional. The result covers nonlinearities that need not be 4-superlinear, including power nonlinearities with exponent in (2,4]. Readers care because the construction works on the whole space under only coercivity of V and does not require stronger growth on f.

Core claim

For the Kirchhoff problem with odd f that is superlinear at infinity with subcritical growth and with continuous coercive V, infinitely many sign-changing solutions are obtained by a combination of the invariant sets method and the Ljusternik-Schnirelmann type minimax method. This holds even when the nonlinearity is not 4-superlinear, such as for the power |u|^{p-2}u with p in (2,4].

What carries the argument

Combination of the invariant sets method and the Ljusternik-Schnirelmann type minimax method applied to the variational functional on the space of sign-changing functions

If this is right

The result applies to power nonlinearities with p in (2,4].
Infinitely many sign-changing solutions exist without 4-superlinearity at infinity.
The coercivity of V guarantees the Palais-Smale condition holds on R^3.
Arbitrarily large numbers of distinct sign-changing solutions can be obtained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same combination of methods may apply to other nonlocal equations with different nonlocal coefficients.
Sign-changing solutions obtained this way may admit further analysis of their nodal sets or symmetry.
The approach could be tested numerically on radial potentials to count the first few solutions explicitly.

Load-bearing premise

The nonlinearity f must be odd in u.

What would settle it

An explicit continuous coercive V and odd superlinear subcritical f for which only finitely many sign-changing solutions exist would disprove the claim.

read the original abstract

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is continuous and coercive. For the case when $f$ is odd in $u$ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity $|u|^{p-2}u$ with $p\in(2,4]$.

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

The paper shows infinitely many sign-changing solutions to a Kirchhoff equation in R^3 for odd nonlinearities that need only be 2-superlinear, using invariant sets plus Ljusternik-Schnirelmann.

read the letter

The main contribution is an existence result for infinitely many sign-changing solutions to the Kirchhoff problem on R^3. When f is odd they make the energy even, apply Ljusternik-Schnirelmann to get a sequence of critical points, and use invariant sets to guarantee those points change sign. The coercivity of V supplies the compact embedding that gives the Palais-Smale condition at each minimax level, and they supply separate estimates to keep (PS) sequences bounded even when the nonlinearity is only p-superlinear for p in (2,4]. That last point is the concrete increment over earlier work, which usually needed 4-superlinearity to close the estimates on the nonlocal term.

Referee Report

0 major / 3 minor

Summary. The manuscript proves existence of infinitely many sign-changing solutions to the Kirchhoff problem -(a + b ∫_{R^3} |∇u|^2) Δu + V(x)u = f(u) in R^3, where a,b>0, V is continuous and coercive, and f is odd with superlinear-at-infinity subcritical growth (including power nonlinearities with p∈(2,4]). The argument applies the invariant-sets technique to the even energy functional and extracts the sign-changing critical points via Ljusternik-Schnirelmann minimax.

Significance. If the technical estimates hold, the result meaningfully extends the literature on sign-changing solutions for nonlocal Kirchhoff equations on R^3 by accommodating nonlinearities that are not necessarily 4-superlinear; the coercivity of V supplies the compact embedding that closes the Palais-Smale argument at every minimax level, while oddness of f guarantees the functional is even.

minor comments (3)

[§1] §1: The comparison with prior existence results for Kirchhoff problems could be sharpened by explicitly stating which earlier works already treat the 4-superlinear case and which do not.
[§3.2] §3.2, Lemma 3.4: The boundedness estimate for (PS) sequences when p≤4 relies on a separate truncation argument; a one-line reference back to the corresponding estimate for the local case would improve readability.
Notation: The symbol c_ε is used both for a generic constant and for a specific sequence; a brief clarification at first appearance would avoid confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The report correctly summarizes the main result: the existence of infinitely many sign-changing solutions to the Kirchhoff problem under the stated assumptions on f and V, obtained via the combination of invariant sets and Ljusternik-Schnirelmann minimax methods. We appreciate the recommendation for minor revision. No specific major comments appear in the report, so we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the invariant-sets technique to the even functional (guaranteed by oddness of f) together with the Ljusternik-Schnirelmann minimax principle on the Nehari manifold or symmetric sets. Coercivity of V supplies the compact embedding for the Palais-Smale condition, while separate estimates bound (PS) sequences for the nonlocal term and sub-4 growth; none of these steps reduces to a fitted parameter, self-definition, or load-bearing self-citation. The methods are standard and externally verifiable, rendering the argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard functional-analytic tools for variational methods on unbounded domains; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Standard Sobolev embeddings and compactness properties hold for the space H^1(R^3) under the coercive potential V.
Invoked implicitly for the variational setting and Palais-Smale condition in R^3.
domain assumption The energy functional satisfies the necessary geometric conditions for the Ljusternik-Schnirelmann minimax theorem on the invariant sets.
Required for the multiplicity argument; location is the method description in the abstract.

pith-pipeline@v0.9.0 · 5744 in / 1362 out tokens · 27169 ms · 2026-05-25T10:05:33.454193+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.1 … combination of invariant sets method and the Ljusternik-Schnirelman type minimax method … A(P±ε) ⊂ P±ε … σ(t,u) … ck := inf sup I
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

I(u) = ½∥u∥²E + (b/4)(∫|∇u|²)² − ∫F(u) … (AR) condition µ>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

38 extracted references · 38 canonical work pages

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L. Li, J.J. Sun and S. Tersian, Inﬁnitely many sign-changing solutions for the Br´ ezis- Nirenberg problem involving the fractional laplacian , Fract. Calc. Appl. Anal., 20 (2017), 1146–1164

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J.J. Sun and C.L. Tang, Resonance problems for Kirchhoﬀ type equations , Discrete Contin. Dyn. Syst., 33 (2013), 2139–2154. SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 23

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J. W ang, L.X. Tian, J.X. Xu and F.B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoﬀ type problem with critical growth , J. Diﬀerential Equations, 253 (2012), 2314–2351

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[1] [1]

Badiale, E

M. Badiale, E. Serra, Semilinear elliptic equations for beginners. Existence re sults via the variational approach, Springer, London, 2011

work page 2011

[2] [2]

Bartsch, Z

T. Bartsch, Z. Liu and T. W eth, Sign changing solutions of superlinear Schr¨ odinger equations, Commun. Partial Diﬀerential Equations, 29 (2004), 25–42

work page 2004

[3] [3]

Bartsch, Z.L

T. Bartsch, Z.L. Liu and T. W eth, Nodal solutions of a p-Laplacian equation, Proc. Lond. Math. Soc., 91 (2005), 129–152

work page 2005

[4] [4]

Bartsch, A

T. Bartsch, A. Pankov and Z.-Q. W ang, Nonlinear Schr¨ odinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 1–21

work page 2001

[5] [5]

Bartsch and Z.-Q

T. Bartsch and Z.-Q. W ang, Existence and multilicity results for some superlinear ell iptic problems on RN , Commun. Partial Diﬀerential Equations, 20 (1995), 1725–1741

work page 1995

[6] [6]

C. Y. Chen, Y. C. Kuo and T. F. W u, The Nehari manifold for a Kirchhoﬀ type problem involving sign-changing weight functions , J. Diﬀerential Equations, 250 (2011), 1876–1908

work page 2011

[7] [7]

Deng, S.J

Y.B. Deng, S.J. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoﬀ-type problems in R3, J. Func. Anal., 269 (2015), 3500–3527. 22 JIJIANG SUN, LIN LI, M. CENCELJ, B. GABROV ˇSEK

work page 2015

[8] [8]

Figueiredo, N

G.M. Figueiredo, N. Ikoma and J.R. Santos J´ unior, Existence and concentration result for the Kirchhoﬀ type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931–979

work page 2014

[9] [9]

He and W.M

X.M. He and W.M. Zou, Existence and concentration behavior of positive solution s for a Kirchhoﬀ type equations in R3, J. Diﬀerential Equations, 252 (2012), 1813–1834

work page 2012

[10] [10]

He and G.B

Y. He and G.B. Li, Standing waves for a class of Kirchhoﬀ type problems in R3 involving critical Sobolev exponents, Calc. Var. Partial Diﬀer. Equ., 54 (2015), 3067–3106

work page 2015

[11] [11]

Huang and Z

Y.S. Huang and Z. Liu, On a class of Kirchhoﬀ type problems, Arch. Math., 102 (2014), 127–139

work page 2014

[12] [12]

Kirchhoﬀ, Mechanik, Teubner, Leipzig, 1883

G. Kirchhoﬀ, Mechanik, Teubner, Leipzig, 1883

work page

[13] [13]

Li and H.Y

G.B. Li and H.Y. Ye, Existence of positive ground state solutions for the nonlin ear Kirchhoﬀ type equations in R3, J. Diﬀerential Equations, 257 (2014), 566–600

work page 2014

[14] [14]

Li and J.J

L. Li and J.J. Sun, Existence and multiplicity of solutions for the Kirchhoﬀ eq uations with asymptotically linear nonlinearities , Nonlinear Anal. Real W orld Appl., 26 (2015), 391–399

work page 2015

[15] [15]

L. Li, J.J. Sun and S. Tersian, Inﬁnitely many sign-changing solutions for the Br´ ezis- Nirenberg problem involving the fractional laplacian , Fract. Calc. Appl. Anal., 20 (2017), 1146–1164

work page 2017

[16] [16]

Y.H. Li, F.Y. Li and J.P. Shi, Existence of a positive solution to Kirchhoﬀ type problems without compactness conditions, J. Diﬀerential Equations, 253 (2012), 2285–2294

work page 2012

[17] [17]

Lions, On some questions in boundary value problem s of mathematical physics, in: Con- temporary Developments in Continuum Mechanics and Partial Diﬀerential Equations, Proc

J.L. Lions, On some questions in boundary value problem s of mathematical physics, in: Con- temporary Developments in Continuum Mechanics and Partial Diﬀerential Equations, Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Ri o de Janeiro, 1977, in: North- Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 19 78, pp. 284–346

work page 1977

[18] [18]

Liu, X.Q

J.Q. Liu, X.Q. Liu and Z.-Q. W ang, Multiple mixed states of nodal solutions for nonlinear Schr¨ oinger systems,Calc. Var. Partial Diﬀer. Equ., 52 (2015), 565–586

work page 2015

[19] [19]

Liu and J.X

Z.L. Liu and J.X. Sun, Invariant sets of descending ﬂow in critical point theory wi th appli- cations to nonlinear diﬀerential equations, J. Diﬀerential Equations, 172, (2001) 257–299

work page 2001

[20] [20]

Z. L. Liu, F. A. van Heerden and Z.-Q. W ang, Nodal type bound states of Schr¨ odinger equa- tions via invariant set and minimax methods, J. Diﬀerential Equations, 214 (2005), 358–390

work page 2005

[21] [21]

Liu, Z.-Q

Z.L. Liu, Z.-Q. W ang and J.J. Zhang, Inﬁnitely many sign-changing solutions for the non- linear Schr¨ odinger-Poisson system,Annali di Matematica, (2015), doi: 10.1007/s10231-015- 0489-8

work page doi:10.1007/s10231-015- 2015

[22] [22]

Liu and S.J

Z.S. Liu and S.J. Guo, Existence and concentration of positive ground states for a Kirchhoﬀ equation involving critical Sobolev exponent, Z. Angew. Math. Phys., 66 (2015), 747–769

work page 2015

[23] [23]

Lu, Signed and sign-changing solutions for a Kirchhoﬀ-type equ ation in bounded do- mains, J

S.S. Lu, Signed and sign-changing solutions for a Kirchhoﬀ-type equ ation in bounded do- mains, J. Math. Anal. Appl., 432 (2015), 965–982

work page 2015

[24] [24]

T. F. Ma and J. E. Mu˜ noz Rivera, Positive solutions for a nonlinear nonlocal elliptic trans - mission problem , Appl. Math. Lett., 16 (2003), 243–248

work page 2003

[25] [25]

A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoﬀ type prob lems without the P.S. condition , Nonlinear Anal., 70 (2009), 1275–1287

work page 2009

[26] [26]

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

D. Naimen, The critical problem of Kirchhoﬀ type elliptic equations in dimension four , J. Diﬀerential Equations, 257 (2014), 1168–1193

work page 2014

[27] [27]

Perera and Z

K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoﬀ-type problems via the Yan g index, J. Diﬀerential Equations, 221 (2006), 246–255

work page 2006

[28] [28]

Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Diﬀerential Equations, CBMS Regional Conference Series in Mathematics, vol

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Diﬀerential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathemat- ical Society, Providence, RI, 1986

work page 1986

[29] [29]

Shuai, Sign-changing solutions for a class of Kirchhoﬀ type proble m in bounded domains , J

W. Shuai, Sign-changing solutions for a class of Kirchhoﬀ type proble m in bounded domains , J. Diﬀerential Equations, 259 (2015), 1256–1274

work page 2015

[30] [30]

Variational Methods. Applications to Nonlinear Partial D iﬀerential Equations and Hamiltonian Systems

M. Struwe, “Variational Methods. Applications to Nonlinear Partial D iﬀerential Equations and Hamiltonian Systems” , 3 rd Edition, Springer–Verlag, Berlin, 2000

work page 2000

[31] [31]

Sun and S.W

J.J. Sun and S.W. Ma, Inﬁnitely many sign-changing solutions for the Br´ ezis-Nirenberg prob- lem, Commun. Pure Appl. Anal., 13 (2014), 2317–2330

work page 2014

[32] [32]

Sun and C.L

J.J. Sun and C.L. Tang, Existence and multiplicity of solutions for Kirchhoﬀ type e quations, Nonlinear Anal., 74 (2011), 1212–1222

work page 2011

[33] [33]

Sun and C.L

J.J. Sun and C.L. Tang, Resonance problems for Kirchhoﬀ type equations , Discrete Contin. Dyn. Syst., 33 (2013), 2139–2154. SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS 23

work page 2013

[34] [34]

W ang, L.X

J. W ang, L.X. Tian, J.X. Xu and F.B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoﬀ type problem with critical growth , J. Diﬀerential Equations, 253 (2012), 2314–2351

work page 2012

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