Opuscula Mathematica
Opuscula Math. 46, no. 5 (2026), 709-750

Normalized solutions of a critical Kirchhoff equation in high dimensional spaces

Ruikang Lu
Qilin Xie
Jianshe Yu
Abstract. The following well-known critical Kirchhoff equation with prescribed mass, \[\begin{cases}\displaystyle{-\Big(a+b\int_{\mathbb{R}^N} |\nabla u|^2dx\Big) \Delta u+\lambda u =\mu |u|^{q-2}u+|u|^{2^*-2}u} &\text{in } \mathbb{R}^N, \; N\geq 4, \\ \displaystyle{\,\int_{\mathbb{R}^N}|u|^2dx=c}, & \end{cases}\] has been extensively studied, where \(a\), \(b\), \(c\), \(\mu\), and \(q\) are constants with \(a,c\gt 0\) and \(2\lt q\lt 2^*\), while \(\lambda\) is a real Lagrange multiplier. When \(\mu=0\), we obtain complete results on the existence, nonexistence, and multiplicity of normalized solutions. When \(\mu\gt 0\) and \(N\geq 5\), we investigate the existence of positive normalized solutions, partly filling a gap in the existing literature. Lastly, when \(\mu\gt 0\) and \(N=4\), we obtain a mountain-pass solution by using a new threshold for the energy functional. The main results of this paper broaden and improve the existing results on normalized solutions of the Kirchhoff equation in high-dimensional spaces.
Keywords: Kirchhoff equation, variational methods, high dimensional spaces, Palais-Smale condition.
Mathematics Subject Classification: 35J60, 47J30, 35J20.
Full text (pdf)
References ▼
  1. [1] M. Al-Gwaiz, V. Benci, F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal. 106 (2014), 18-34. https://doi.org/10.1016/j.na.2014.04.011
  2. [2] T. Bartsch, N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations 58 (2019) 22, 1-24. https://doi.org/10.1007/s00526-018-1476-x
  3. [3] H. Brézis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), no. 2, 137-151.
  4. [4] P.C. Carrião, O.H. Miyagaki, A. Vicente, Normalized solutions of Kirchhoff equations with critical and subcritical nonlinearities: the defocusing case, Partial Differ. Equ. Appl. 3 (2022), no. 5, 64. https://doi.org/10.1007/s42985-022-00201-3
  5. [5] C.-Y. Chen, Y.-C. Kuo, T.-F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), no. 4, 1876-1908. https://doi.org/10.1016/j.jde.2010.11.017
  6. [6] S. Chen, X. Tang, Another look at Schrödinger equations with prescribed mass, J. Differential Equations 386 (2024), 435-479. https://doi.org/10.1016/j.jde.2023.12.026
  7. [7] S. Chen, X. Tang, Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities, Math. Ann. 391 (2025), no. 2, 2783-2836. https://doi.org/10.1007/s00208-024-02982-x
  8. [8] M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), no. 7, 4619-4627. https://doi.org/10.1016/s0362-546x(97)00169-7
  9. [9] F.J.S.A. Corrêa, D.C. de Morais Filho, On a class of nonlocal elliptic problems via Galerkin method, J. Math. Anal. Appl. 310 (2005), no. 1, 177-187. https://doi.org/10.1016/j.jmaa.2005.01.052
  10. [10] X. Feng, H. Liu, Z. Zhang, Normalized solutions for Kirchhoff type equations with combined nonlinearities: the Sobolev critical case, Discrete Contin. Dyn. Syst. 43 (2023), no. 8, 2935-2972. https://doi.org/10.3934/dcds.2023035
  11. [11] X. Fang, Z. Ou, Y. Lv, Normalized solutions to the Kirchhoff equation with triple critical exponents in \(\mathbb{R}^4\), Appl. Math. Lett. 153 (2024), 109062. https://doi.org/10.1016/j.aml.2024.109062
  12. [12] G.M. Figueiredo, N. Ikoma, J.R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal. 213 (2014), no. 3, 931-979. https://doi.org/10.1007/s00205-014-0747-8
  13. [13] H. Guo, Y. Zhang, H. Zhou, Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential, Commun. Pure Appl. Anal. 17 (2018), no. 5, 1875-1897. https://doi.org/10.3934/cpaa.2018089
  14. [14] X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\), J. Differential Equations 252 (2012), no. 2, 1813-1834. https://doi.org/10.1016/j.jde.2011.08.035
  15. [15] J. Hu, A. Mao, Normalized solutions to the Kirchhoff equation with a perturbation term, Differ. Integral Equ. 36 (2023), no. 3-4, 289-312. https://doi.org/10.57262/die036-0304-289
  16. [16] L. Jeanjean, J. Jendrej, T.T. Le, N. Visciglia, Orbital stability of ground states for a Sobolev critical Schrödinger equation, J. Math. Pures Appl. 164 (2022), 158-179. https://doi.org/10.1016/j.matpur.2022.06.005
  17. [17] L. Jeanjean, T.T. Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann. 384 (2022), no. 1-2, 101-134. https://doi.org/10.1007/s00208-021-02228-0
  18. [18] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
  19. [19] L. Kong, H. Chen, Normalized ground states for the mass-energy doubly critical Kirchhoff equations, Acta Appl. Math. 186 (2023), 5. https://doi.org/10.1007/s10440-023-00584-4
  20. [20] L. Kong, L. Zhu, H. Chen, Two positive normalized solutions for the mass-energy doubly critical Kirchhoff equation, J. Fixed Point Theory Appl. 27, 101 (2025). https://doi.org/10.1007/s11784-025-01251-x
  21. [21] G. Li, X. Luo, T. Yang, Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent, Ann. Fenn. Math. 47 (2022), no. 2, 895-925. https://doi.org/10.54330/afm.120247
  22. [22] Y. Li, X. Hao, J. Shi, The existence of constrained minimizers for a class of nonlinear Kirchhoff-Schrödinger equations with doubly critical exponents in dimension four, Nonlinear Anal. 186 (2019), 99-112. https://doi.org/10.1016/j.na.2018.12.010
  23. [23] E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/014
  24. [24] J.-L. Lions, On some questions in boundary value problems of mathematical physics, [in:] Contemporary Developments in Continuum Mechanics and Partial Differential Equations, North-Holland Math. Stud., vol. 30, Elsevier, Amsterdam, 1978, 284-346. https://doi.org/10.1016/s0304-0208(08)70870-3
  25. [25] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. https://doi.org/10.1007/978-1-4612-5282-5
  26. [26] S. Qi, W. Zou, Exact number of positive solutions for the Kirchhoff equation, SIAM J. Math. Anal. 54 (2022), no. 5, 5424-5446. https://doi.org/10.1137/21m1445879
  27. [27] V.D. Rădulescu, Kirchhoff Equations: A Variational Approach, 1st ed., Chapman and Hall/CRC. https://doi.org/10.1201/9781003797159
  28. [28] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations 269 (2020), no. 9, 6941-6987. https://doi.org/10.1016/j.jde.2020.05.016
  29. [29] N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal. 279 (2020), no. 6, 108610. https://doi.org/10.1016/j.jfa.2020.108610
  30. [30] J. Wei, Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal. 283 (2022), no. 6, 109574. https://doi.org/10.1016/j.jfa.2022.109574
  31. [31] Q. Xie, S. Ma, X. Zhang, Bound state solutions of Kirchhoff type problems with critical exponent, J. Differential Equations 261 (2016), no. 2, 890-924. https://doi.org/10.1016/j.jde.2016.03.028
  32. [32] Q. Xie, X. Wu, C. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2773-2786. https://doi.org/10.3934/cpaa.2013.12.2773
  33. [33] Q. Xie, B.-X. Zhou, A study on the critical Kirchhoff problem in high-dimensional space, Z. Angew. Math. Phys. 73 (2022), no. 1, Paper No. 4, 29. https://doi.org/10.1007/s00033-021-01626-3
  34. [34] L. Xu, Q. Xie, The complementarity of normalized solutions for Kirchhoff equations with mixed nonlinearity, J. Math. Res. Appl. 44 (2024), no. 4, 483-494. https://doi.org/10.3770/j.issn:2095-2651.2024.04.007
  35. [35] L. Xu, F. Li, Q. Xie, Existence and multiplicity of normalized solutions with positive energy for the Kirchhoff equation, Qual. Theory Dyn. Syst. 23 (2024), no. 3, Paper No. 135, 23. https://doi.org/10.1007/s12346-024-01001-3
  36. [36] L. Xu, C. Song, Q. Xie, Multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearity, Taiwanese J. Math. 28 (2024), no. 3, 589-609. https://doi.org/10.11650/tjm/240202
  37. [37] H. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci. 38 (2015), no. 13, 2663-2679. https://doi.org/10.1002/mma.3247
  38. [38] H. Ye, The existence of normalized solutions for \(L^2\)-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 66 (2015), no. 4, 1483-1497. https://doi.org/10.1007/s00033-014-0474-x
  39. [39] H. Ye, The mass concentration phenomenon for \(L^2\)-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 67 (2016), 1-16. https://doi.org/10.1007/s00033-016-0624-4
  40. [40] X. Zeng, Y. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett. 74 (2017), 52-59. https://doi.org/10.1016/j.aml.2017.05.012
  41. [41] J. Zhang, J. Zhang, X. Zhong, Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity, arXiv preprint arXiv:2210.12911 (2022).
  42. [42] J. Zhang, X. Zhong, J. Zhou, Normalized solutions of coupled Sobolev critical Schrödinger equations with mass subcritical couplings, Bull. Math. Sci. (2025), 2550027. https://doi.org/10.1142/S1664360725500274
About the authors ▼
Ruikang Lu
Guangdong University of Technology, School of Mathematics and Statistics, Guangzhou, Guangdong 510520, P.R. China
Qilin Xie (corresponding author)
https://orcid.org/0000-0002-6569-5005
Guangdong University of Technology, School of Mathematics and Statistics, Guangzhou, Guangdong 510520, P.R. China
Jianshe Yu
Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou, Guangdong 510006, P.R. China
About the article ▼
Communicated by Vicenţiu D. Rădulescu.
Received: 2026-08-05.
Revised: 2026-10-07.
Accepted: 2026-10-08.
Published online: 2026-10-28.
Cite this article as ▼
Ruikang Lu, Qilin Xie, Jianshe Yu, Normalized solutions of a critical Kirchhoff equation in high dimensional spaces, Opuscula Math. 46, no. 5 (2026), 709-750, https://doi.org/10.7494/OpMath.202610082.
Download this article's citation as: