pith. sign in

arxiv: 1907.03200 · v2 · pith:OLSISLFYnew · submitted 2019-07-06 · 🧮 math.DS · math-ph· math.MP

Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in mathbb{R}^(N)

Dong-Lun Wu , Fengying Li This is my paper

Pith reviewed 2026-05-25 01:14 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MP
keywords Kirchhoff equationsfourth-order elliptic equationsconcave-convex nonlinearitiesvariational methodsexistence and multiplicitymountain pass theoremgenus theory
0
0 comments X

The pith

Fourth-order Kirchhoff elliptic equations in R^N have at least one solution for λ=0, two for small λ>0, and infinitely many if the nonlinearity is odd.

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

The paper uses variational methods to prove existence and multiplicity of solutions for a fourth-order Kirchhoff-type equation with a parameter λ multiplying a sublinear term and a superlinear term. For λ equal to zero the equation has at least one solution. When λ is positive but small there are at least two nontrivial solutions. If the full nonlinearity is odd then infinitely many solutions exist regardless of λ. These results extend earlier findings on such nonlocal problems.

Core claim

We show the existence of at least one solution for the equation when λ=0. For λ>0 small enough, at least two nontrivial solutions are obtained. If f(x,u) is odd in u, the equation possesses infinitely many solutions for all λ≥0. The proofs rely on the mountain-pass theorem and genus theory applied to the energy functional.

What carries the argument

The associated energy functional on a suitable Sobolev space, whose critical points correspond to weak solutions of the equation, analyzed via mountain-pass and symmetric mountain-pass theorems.

If this is right

  • The equation has a ground state solution when λ=0.
  • Adding a small concave term creates an additional local minimum solution.
  • Odd symmetry allows application of genus theory to produce an unbounded sequence of critical values.
  • The results hold under general 3-superlinear conditions on the convex part.

Where Pith is reading between the lines

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

  • The approach may extend to equations with different nonlocal terms or in bounded domains.
  • Similar multiplicity could hold for other parameter ranges if additional assumptions are imposed.
  • Numerical simulations could confirm the predicted number of solutions for concrete choices of M and V.

Load-bearing premise

The Kirchhoff function M and the nonlinear terms k and h must satisfy specific growth and regularity conditions that ensure the energy functional is well-defined and satisfies the geometric hypotheses of the critical point theorems used.

What would settle it

Constructing explicit functions M, V, k, h that meet all stated hypotheses but for which the energy functional has no critical points of the expected type would disprove the claims.

read the original abstract

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N}, \end{eqnarray*} where $M(t):\mathbb{R}\rightarrow\mathbb{R}$ is the Kirchhoff function, $f(x,u)=\lambda k(x,u)+ h(x,u)$, $\lambda\geq0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $\lambda=0$. For $\lambda>0$ small enough, we obtain at least two nontrivial solutions. Furthermore, if $f(x,u)$ is odd in $u$, we show that above equations possess infinitely many solutions for all $\lambda\geq0$. Our theorems generalize some known results in the literatures even for $\lambda=0$ and our proof is based on the variational methods.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes existence and multiplicity of solutions to the fourth-order Kirchhoff-type equation Δ²u − M(‖∇u‖₂²)Δu + V(x)u = λk(x,u) + h(x,u) on ℝ^N. It claims at least one solution for λ=0, at least two nontrivial solutions for sufficiently small λ>0, and infinitely many solutions (via genus theory) when f is odd in u, for all λ≥0. The proofs rely on variational methods applied to the associated energy functional, under growth and sign conditions on M, V, k, and h that ensure the functional is C¹, satisfies mountain-pass geometry, and obeys the Palais-Smale condition.

Significance. If the stated hypotheses on M, V, k, and h are verified to make the functional well-defined on the appropriate Sobolev space and to recover the required geometry and compactness, the results provide a natural extension of concave-convex multiplicity theorems to the biharmonic Kirchhoff setting on unbounded domains. The approach is standard but the combination of the nonlocal Kirchhoff term with the fourth-order operator and the mixed sub/superlinear nonlinearity adds technical interest; explicit comparison to prior results (even for λ=0) would strengthen the contribution.

minor comments (3)
  1. [Abstract] Abstract: the claim that the theorems 'generalize some known results in the literatures even for λ=0' should be supported by at least one or two explicit citations and a brief comparison of the hypotheses on M and the growth of h.
  2. [Introduction / equation (1.1)] The notation for the Kirchhoff term M(‖∇u‖₂²)Δu in the equation should be cross-checked against the integrated form (1/2)∫₀^{‖∇u‖²} M(s) ds that appears in the energy functional; any discrepancy in the definition of M should be clarified in the introduction.
  3. [Introduction] The abstract invokes 'standard variational methods' without naming the precise theorems (mountain-pass, symmetric mountain-pass, or genus); a short paragraph in the introduction listing the critical-point results employed would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We appreciate the suggestion regarding explicit comparisons to prior results and will address it accordingly.

read point-by-point responses

  1. Referee: explicit comparison to prior results (even for λ=0) would strengthen the contribution.

    Authors: We agree that a more detailed comparison would improve clarity. In the revised manuscript, we will expand the introduction to include explicit discussions of how our theorems extend existing results on fourth-order Kirchhoff equations with concave-convex nonlinearities, particularly highlighting the novelties even in the case λ=0 relative to works such as those on biharmonic problems and Kirchhoff-type equations in the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard application of external critical-point theorems

full rationale

The derivation applies the mountain-pass theorem and symmetric mountain-pass/genus theory to an energy functional built directly from the PDE under stated growth/sign conditions on M, V, k, h. These are independent external theorems whose hypotheses are verified in the paper; no quantity is fitted inside the paper and then renamed a prediction, no self-citation supplies a uniqueness theorem or ansatz, and the existence statements do not reduce to the inputs by definition. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on domain assumptions from nonlinear analysis that are standard but not independently verified here.

axioms (1)
  • domain assumption The energy functional satisfies the Palais-Smale condition and mountain-pass geometry under the stated growth conditions on M, V, k, h.
    Invoked to obtain critical points corresponding to solutions.

pith-pipeline@v0.9.0 · 5735 in / 1142 out tokens · 31780 ms · 2026-05-25T01:14:12.713348+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

  1. [1]

    Ansari and S.M

    H. Ansari and S.M. Vaezpour, Existence and multiplicity of solution s for fourth-order elliptic Kirch- hoff equations with potential term. Complex Var. Elliptic Equ. 60 (2015)(5), 668-695

    work page 2015

  2. [2]

    Bartsch, and Z.Q

    T. Bartsch, and Z.Q. Wang, Existence and multiplicity results for s ome superlinear elliptic problems on RN . Comm. Partial Diff. Equ. 20 (1995) (9-10), 1725-1741

    work page 1995

  3. [3]

    X. Cao, J. Xu, J. Wang, Multiple positive solutions for Kirchhoff typ e problems involving concave and convex nonlinearities in R3. Electron. J. Qual. Theory Differ. Equ. 2016, Paper No. 301, 16 pp

    work page 2016

  4. [4]

    Chang, Infinite dimensional Morse theory and multiple solution problems, Birkh¨ auser, 1993

    K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkh¨ auser, 1993

    work page 1993

  5. [5]

    C. Chen, J. Huang, L. Liu, Multiple solutions to the nonhomogeneo us p-Kirchhoff elliptic equation with concave-convex nonlinearities. Appl. Math. Lett. 26 (2013)(7), 754-759

    work page 2013

  6. [6]

    C. Chen, Q. Chen, Infinitely many solutions for p-Kirchhoff equat ion with concave-convex nonlin- earities in RN . Math. Methods Appl. Sci. 39 (2016)(6), 1493-1504

    work page 2016

  7. [7]

    Chu, J.-J

    C.-M. Chu, J.-J. Sun, Z.-P. Cai, Multiple solutions for a Kirchhoff-ty pe problem involving nonlocal fractional p-Laplacian and concave-convex nonlinearities. Rocky Mountain J. Math. 47 (2017)(6), 1803-1823

    work page 2017

  8. [8]

    Ding and L

    L. Ding and L. Li, Two nontrivial solutions for the nonhomogenous fourth order Kirchhoff equation. Z. Anal. Anwend. 36 (2017)(2), 191-207

    work page 2017

  9. [9]

    Ferrara, S

    M. Ferrara, S. Khademloo and S. Heidarkhani, Multiplicity results f or perturbed fourth-order Kirch- hoff type elliptic problems. Appl. Math. Comput. 234 (2014), 316-325

    work page 2014

  10. [10]

    Huang, C

    J. Huang, C. Chen, Z. Xiu, Existence and multiplicity results for a p-Kirchhoff equation with a concave-convex term. Appl. Math. Lett. 26 (2013)(11), 1070-1075

    work page 2013

  11. [11]

    Heidarkhani, M

    S. Heidarkhani, M. Ferrara and S. Khademloo, Nontrivial solutio ns for one-dimensional fourth-order Kirchhoff-type equations. Mediterr. J. Math. 13 (2016)(1), 217-236

    work page 2016

  12. [12]

    Lions, On some questions in boundary value problems of math ematical physics

    J.L. Lions, On some questions in boundary value problems of math ematical physics. In Proceedings of International Symposium on Continuum Mechanics and Part ial Differential Equations (North- Holland Mathematics Studies, North-Holland, Amsterdam/New York , 1978), Vol. 30, pp. 284-346. 12

    work page 1978

  13. [13]

    H.Y. Li, J.F. Liao, Existence and multiplicity of solutions for a super linear Kirchhoff-type equations with critical Sobolev exponent in RN . Comput. Math. Appl. 72 (2016) 2900-2907

    work page 2016

  14. [14]

    Liao, H.Y

    J.F. Liao, H.Y. Li, P. Zhang, Existence and multiplicity of solutions f or a nonlocal problem with critical Sobolev exponent. Comput. Math. Appl. 75 (2018) 787-797

    work page 2018

  15. [15]

    Li and W.-W

    L. Li and W.-W. Pan, A note on nonlinear fourth-order elliptic equ ations on RN . J. Global Optim. 57 (2013)(4), 1319-1325

    work page 2013

  16. [16]

    Song and C.S

    H.X. Song and C.S. Chen, Infinitely Many Solutions for Schr¨ oding er-Kirchhoff-Type Fourth-Order Elliptic Equations. Proc. Edinb. Math. Soc. 60 (2017)(4), 1003-1020

    work page 2017

  17. [17]

    J. Sun, L. Li, M. Cencelj, B. Gabrov˘ sek, Infinitely many sign-c hanging solutions for Kirchhoff type problems in R3, Nonlinear Anal. (2018), https://doi.org/10.1016/j.na.2018.10.007

    work page doi:10.1016/j.na.2018.10.007 2018

  18. [18]

    Tang, Infinitely many solutions for semilinear Schr¨ odinger e quations with sign-changing po- tential and nonlinearity

    X.H. Tang, Infinitely many solutions for semilinear Schr¨ odinger e quations with sign-changing po- tential and nonlinearity. J. Math. Anal. Appl. 401 (2013), 407-15

    work page 2013

  19. [19]

    Wang and Y

    F. Wang and Y. An, Existence and multiplicity of solutions for a fou rth-order elliptic equation. Bound. Value Probl. 2012, 2012:6, 9 pp

    work page 2012

  20. [20]

    F. Wang, M. Avci and Y. An, Existence of solutions for fourth o rder elliptic equations of Kirchhoff type. J. Math. Anal. Appl. 409 (2014)(1), 140-146

    work page 2014

  21. [21]

    D.-L. Wu, C. Li, P.F. Yuan, Multiplicity solutions for a class of fract ional Hamiltonian systems with concave-convex potentials. Mediterr. J. Math. 15(2018), 35

    work page 2018

  22. [22]

    Wu, C.-L

    D.-L. Wu, C.-L. Tang, X.-P. Wu, Homoclinic orbits for a class of sec ond-order Hamiltonian systems with concave-convex nonlinearities. Electron. J. Qual. Theory Differ. Equ. 6 (2018), 1-18

    work page 2018

  23. [23]

    D.-L. Wu, X. Yu, New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity, arXiv preprint arXiv:1906.00461 (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  24. [24]

    Xiang, B

    M. Xiang, B. Zhang, M. Ferrara, Multiplicity results for the non- homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities. Proc. A. 471 (2015)(2177), 20150034, 14 pp

    work page 2015

  25. [25]

    Xu and H

    L. Xu and H. Chen, Existence and multiplicity of solutions for four th-order elliptic equations of Kirchhoff type via genus theory. Bound. Value Probl. 2014, 2014:212, 12 pp

    work page 2014

  26. [26]

    Xu and H

    L. Xu and H. Chen, Multiplicity results for fourth order elliptic equ ations of Kirchhoff-type. Acta Math. Sci. Ser. B 35 (2015)(5), 1067-1076

    work page 2015

  27. [27]

    L. Yang, T. An, Existence of multiple solutions for fractional p- Kirchhoff equations with concave- convex nonlinearities. Bound. Value Probl. 2017, Paper No. 27, 12 pp

    work page 2017

  28. [28]

    R. Yuan, Z. Zhang, Homoclinic solutions for a class of second ord er Hamiltonian systems. Results in Math. 61 (2012), 195-208

    work page 2012

  29. [29]

    Zhang, Q

    Q.Y. Zhang, Q. Wang, Multiple solutions for a class of sublinear Sch r¨ odinger equationsJ. Math. Anal. Appl. 389 (2012), 511-518

    work page 2012

  30. [30]

    F. Zhou, K. Wu and X. Wu, High energy solutions of systems of Kir chhoff-type equations on RN . Comput. Math. Appl. 66 (2013)(7), 1299-1305. 13

    work page 2013