Opuscula Mathematica
Opuscula Math. 46, no. 4 (2026), 541-562

Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain

Shupei Shen
Xiaoming He
Abstract. In this paper, we study the existence of the solutions for the following \(p\)-Kirchhoff equation on bounded domain \[\begin{cases}-\Bigg(a+b\displaystyle\int_{\Omega}|\nabla u|^pdx\Bigg)\Delta_pu+\lambda|u|^{p-2}u=h(u), &x \in \Omega,\\ u=0, &x\in\partial\Omega,\end{cases}\] with prescribed mass \[\int_{\Omega}|u|^pdx=m^p,\] where \(2\leq p\lt 3\), \(a\gt 0\), \(b\gt 0\) are positive constants, \(h(u)\) is a general nonlinearity with Sobolev subcritical growth, \(\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(\Omega\subset\mathbb{R}^3\) is a bounded domain, \(\lambda\in \mathbb{R}\) appears as a Lagrange multiplier. First, we prove that the equation admits a positive solution which is a local minimizer when \(h\) is \(L^p\)-subcritical or \(L^p\)-critical at infinity by Brezis-Nirenberg technique. Moreover, we get the multiplicity result by using the genus theory. Next, we prove that the equation has a local minimizer when \(h\) is \(L^p\)-supercritical. Besides, by using the Pohozaev analysis, we prove that in this case, if \(\Omega\) is a star-shaped domain with respect to the origin, the equation admits a second solution which is mountain pass type. To the best of our knowledge, this work seems to be the first contribution on the existence of normalized solutions for \(p\)-Kirchhoff equation on bounded domain.
Keywords: \(p\)-Laplacian Kirchhoff equation, normalized solutions, existence and multiplicity, bounded domain.
Mathematics Subject Classification: 35A15, 35B33, 35J20, 35J60.
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References ▼
  1. [1] C.O. Alves, F.J.S.A. Corrêa, T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85-93. https://doi.org/10.1016/j.camwa.2005.01.008
  2. [2] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305-330. https://doi.org/10.1090/s0002-9947-96-01532-2
  3. [3] J. Borthwick, X. Chang, L. Jeanjean, N. Soave, Bounded Palais-Smale sequences with Morst type information for some constrained functionals, arXiv:2210.12626v2 [math.AP].
  4. [4] L. Cai, P. Ji, P. Jin, Concentration of solutions with prescribed mass for \((p, q)\)-Laplacian equations with nonlocal reaction, Bull. Math. Sci. 16 (2026), 2550024. https://doi.org/10.1142/s1664360725500249
  5. [5] S. Chen, X. Tang, Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities, Math. Ann. 391 (2025), 2783-2836. https://doi.org/10.1007/s00208-024-02982-x
  6. [6] G.M. Figueiredo, N. Ikoma, J.R. Santos, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal. 213 (2014), 931-979. https://doi.org/10.1007/s00205-014-0747-8
  7. [7] F. Gao, Y. Guo, Normalized solutions for \(p\)-Laplacian equation with general nonlinearities on bounded domain, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2025), 1-31. https://doi.org/10.1017/prm.2025.10098
  8. [8] X. He, W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin., Engl. Ser. 26 (2010), 387-394. https://doi.org/10.1007/s10255-010-0005-2
  9. [9] X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\), Journal of Differential Equations 252 (2012), 1813-1834. https://doi.org/10.1016/j.jde.2011.08.035
  10. [10] Z. Jin, W. Zhang, Existence and nonexistence of normalized solutions for nonlinear Schrödinger equation involving combined nonlinearities in bounded domain, arXiv:2602.16263 [math.AP]. https://doi.org/10.1016/j.na.2024.113581
  11. [11] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \(\mathbb{R}^N\), Proc. Roy. Soc. Edinb. A. 129 (1999), 787-809. https://doi.org/10.1017/s0308210500013147
  12. [12] 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
  13. [13] L. Jeanjean, J. Zhang, X. Zhong, A global branch approach to normalized solutions for the Schrödinger equation, J. Math. Pures Appl. 183 (2024), 44-75. https://doi.org/10.1016/j.matpur.2024.01.004
  14. [14] L. Jeanjean, T.T. Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann. 384 (2022), 101-134. https://doi.org/10.1007/s00208-021-02228-0
  15. [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. https://doi.org/10.1007/bf01693821
  16. [16] G. Li, X. Luo, T. Yang, Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent, Ann. Fenn. Math. 47 (2022), 895-925. https://doi.org/10.54330/afm.120247
  17. [17] Q. Li, J. Nie, W. Zhang, Existence and asymptotics of normalized ground states for a Sobolev critical Kirchhoff equation, J. Geom. Anal. 33 (2023), 126. https://doi.org/10.1007/s12220-022-01171-z
  18. [18] X. Li, Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities, Calc. Var. Partial Differ. Equ. 60 (2021), 169. https://doi.org/10.1007/s00526-021-02020-7.
  19. [19] H. Liu, V.D. Rădulescu, Normalized solutions of quasilinear problems with nonlocal reaction, Bull. Math. Sci. (2026), 2650006. https://doi.org/10.1142/s1664360726500062
  20. [20] Y. Liu, L. Zhao, Normalized solutions for Schrödinger equations with general nonlinearities on bounded domains, J. Geom. Anal. 35 (2025), 54. https://doi.org/10.1007/s12220-024-01890-5
  21. [21] B. Noris, H. Tavares, G. Verzini, Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity 32 (2019), 1044-1072. https://doi.org/10.1088/1361-6544/aaf2e0
  22. [22] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equations 221 (2006), 246-255. https://doi.org/10.1016/j.jde.2005.03.006
  23. [23] D. Pierotti, G. Verzini, J. Yu, Normalized solutions for Sobolev critical {Schrödinger} equations on bounded domains, SIAM J. Math. Anal. 57 (2025), 262-285. https://doi.org/10.1137/24m1656281
  24. [24] S. Qi, W. Zou, Normalized solutions of nonhomogeneous mass supercritical Schrödinger equations in bounded domains, J. Geom. Anal. 34 (2024), 59. https://doi.org/10.1007/s12220-023-01504-6
  25. [25] Z. Ren, Y. Lan, Normalized solution to \(p\)-Kirchhoff-type equation in \(\mathbb{R}^N\), Anal. Math. Phys. 14 (2024), 93. https://doi.org/10.1007/s13324-024-00954-7
  26. [26] Z. Ren, Y. Lan, Normalized solution for \(p\)-Kirchhoff equation with a \(L^2\)-supercritical growth, Acta Math. Appl. Sinica, Eng. Ser. 40 (2024), 414-429. https://doi.org/10.1007/s10255-024-1120-9
  27. [27] L. Song, W. Zou, On multiple sign-changing normalized solutions to the Brezis-Nirenberg problem, Math. Z. 309 (2025), 61. https://doi.org/10.1007/s00209-025-03698-7
  28. [28] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamilton Systems, 4th ed., Springer-Verlag, 2008. https://doi.org/10.1007/978-3-540-74013-1
  29. [29] Q. Wang, X. Chang, Normalized solutions of \(L^2\)-supercritical Kirchhoff equations in bounded domains, J. Geom. Anal. 34 (2024), 358. https://doi.org/10.1007/s12220-024-01793-5
  30. [30] R. Wang, S. Yao, J. Sun, Normalized Solutions of Mass-Subcritical Kirchhoff Equations with Potential on Bounded Domains, Mediterr. J. Math. 23 (2026), No. 16, 2026. https://doi.org/10.1007/s00009-025-03002-1
  31. [31] M. Willem, Minimax Theorems, Progr. Nonlinear Differ. Equ. Appl., vol. 24, Birkhäuser, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
  32. [32] W. Xi, H. Chen, Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems, Comput. Math. Appl. 76 (2018), 579-591. https://doi.org/10.1016/j.camwa.2018.04.038
  33. [33] H. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci. 38 (2015), 2663-2679. https://doi.org/10.1002/mma.3247
  34. [34] H. Ye, The mass concentration phenomenon for \(L^2\)-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys. 67 (2016), 29. https://doi.org/10.1007/s00033-016-0624-4
  35. [35] P. Zhang, Normalized solutions for Kirchhoff equations on bounded domains with Sobolev critical or subcritical exponent, Z. Angew. Math. Phys. 76 (2025), 120. https://doi.org/10.1007/s00033-025-02502-0
  36. [36] P. Zhang, Z. Han, Normalized ground states for Kirchhoff equations in \(\mathbb{R}^N\) with a critical nonlinearity, J. Math. Phys. 63 (2022), 021505. https://doi.org/10.1063/5.0067520
  37. [37] Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006), 456-463. https://doi.org/10.1016/j.jmaa.2005.06.102
  38. [38] Z. Zhang, Z. Zhang, Normalized solutions to \(p\)-Laplacian equations with combined nonlinearities, Nonlinearity 35 (2022), 5621-5663. https://doi.org/10.1088/1361-6544/ac902c
  39. [39] J. Zhou, P. Yang, Y. Yu, Normalized solutions for the \(p\)-Laplacian equations with potentials and general nonlinearities, J. Math. Anal. Appl. 558 (2026), 130393. https://doi.org/10.1016/j.jmaa.2026.130393
  40. [40] J. Zhou, P. Yang, Y. Yu, Multiplicity and asymptotic behavior of normalized solutions to \(p\)-Kirchhoff equations, arXiv:2411.11037 [math.AP].
About the authors ▼
Shupei Shen
https://orcid.org/0009-0005-1465-5133
Minzu University of China, College of Science, Zhongguancun South Street 27, Beijing, China
Xiaoming He (corresponding author)
Minzu University of China, College of Science, Zhongguancun South Street 27, Beijing, China
About the article ▼
Communicated by Wen Zhang.
Received: 2026-04-15.
Revised: 2026-06-01.
Accepted: 2026-06-19.
Published online: 2026-08-04.
Cite this article as ▼
Shupei Shen, Xiaoming He, Normalized solutions for p-Kirchhoff equation with general nonlinearities on bounded domain, Opuscula Math. 46, no. 4 (2026), 541-562, https://doi.org/10.7494/OpMath.202606191.
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