Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations

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Abstract

In this paper, we explored the following p-Kirchhoff equation with the L^p-mass constraint

$\left\{\begin{array}{l} -\left(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{p} d x\right) \Delta_{p} u=\lambda|u|^{p-2} u+|u|^{q-2} u, x \in \mathbb{R}^{3} \\ \left(\int_{\mathbb{R}^{3}}|u|^{p} d x\right)^{\frac{1}{p}}=c>0, \end{array}\right.$

where $a>0, b>0, \frac{3}{2}<p<3, p<q<p^{*}:=\frac{3 p}{3-p}, \Delta_{p} u=\operatorname{div}\left(|\nabla u|^{p-2} \nabla u\right)$, and $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier. We considered L^p-mass subcritical, L^p-mass critical, and L^p-mass supercritical cases. Making full use of the Gagliardo-Nirenberg inequality, we established the existence of the minimizers in L^p-mass subcritical and L^p-mass critical cases and the nonexistence in the L^p-mass critical case. For the L^p-mass supercritical case, a radially symmetric ground state was obtained using the Nehari-Pohozaev manifold method. On the other hand, employing a σ-homotopy stable argument of fountain theorem type, we obtained infinitely many radial normalized solutions. In addition, the asymptotic behavior of normalized solutions as $b \rightarrow 0^{+}$ was also explored. Finally, we proved the uniqueness of normalized solutions and provided precise descriptions when $\frac{3}{2}<p \leq 2$.

Keywords

p-Kirchhoff equation / normalized solution / ground state / multiplicity / asymptotic behavior

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References

[1] W. Wang, Q. Li, J. Zhou, Y. Li, Normalized solutions for p-Laplacian equations with a L²-supercritical growth , Ann. Funct. Anal., 12 (2021), 9. https://doi.org/10.1007/s43034-020-00101-w

[2] 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

[3] Q. Lou, Z. Zhang, Multiplicity and concentration of normalized solutions to p-laplacian equations , Z. Angew. Math. Phys., 75 (2024), 81. https://doi.org/10.1007/s00033-024-02219-6

[4] N. S. Papageorgiou, V. D. Rădulescu, W. Zhang, Nonlinear double phase eigenvalue problems, Math. Mech. Solids, 31 (2026), 580-595. https://doi.org/10.1177/10812865251397537

[5] J. Zhang, W. Zhang, V. D. Rădulescu, Localized concentration of semiclassical solutions for double phase problems with nonlocal reaction, Anal. Math. Phys., 16 (2026), 35. https://doi.org/10.1007/s13324-026-01182-x

[6] G. R. Kirchhoff, Vorlesungen über Mathematische Physik. Mechanik , Leipzig, 1876.

[7] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. https://doi.org/10.1016/S0304-0208(08)70870-3

[8] C. Chu, J. Liu, Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation , Adv. Nonlinear Anal., 13 (2024), 20240018. https://doi.org/10.1515/anona-2024-0018

[9] F. J. S. A. Corrêa, G. M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii’s genus , Appl. Math. Lett., 22 (2009), 819-822. https://doi.org/10.1016/j.aml.2008.06.042

[10] M. F. Furtado, L. D. De Oliveira, J. P. P. Da Silva, Multiple solutions for a Kirchhoff equation with critical growth, Z. Angew. Math. Phys., 70 (2019), 11. https://doi.org/10.1007/s00033-018-1045-3

[11] J. Yang, H. Chen, Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth, Adv. Nonlinear Anal., 14 (2025), 20250075. https://doi.org/10.1515/anona-2025-0075

[12] 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

[13] Z. Wang, H. Sun, Normalized solutions for Kirchhoff equations with Choquard nonlinearity: mass super-critical case, Commun. Anal. Mech., 17 (2025), 317-340. https://doi.org/10.3934/cam.2025013

[14] 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

[15] Y. Ni, J. Sun, J. Chen, Multiplicity, concentration of normalized solutions for a kirchhoff type problem with L²-subcritical nonlinearities , Commun. Anal. Mech., 16 (2024), 633-654. https://doi.org/10.3934/cam.2024029

[16] Z. Ren, Y. Lan, Normalized solution for p-Kirchhoff equation with a L²-supercritical growth , Acta Math. Appl. Sin. Engl. Ser., 40 (2024), 414-429. https://doi.org/10.1007/s10255-024-1120-9

[17] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. Theory Methods Appl., 28 (1997), 1633-1659. https://doi.org/10.1016/s0362-546x(96)00021-1

[18] J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-924. https://doi.org/10.1512/iumj.2000.49.1893

[19] M. Agueh, Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations , Nonlinear Differ. Equations Appl., 15 (2008), 457-472. https://doi.org/10.1007/s00030-008-7021-4

[20] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576. https://doi.org/10.1007/BF01208265

[21] L. Baldelli, T. Yang, Normalized solutions to a class of (2, q)-Laplacian equations , Adv. Nonlinear Stud., 25 (2025), 225-256. https://doi.org/10.1515/ans-2023-0163

[22] H. Berestycki, P. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. https://doi.org/10.1007/BF00250556

[23] M. Willem, Minimax Theorems, 1^st edition, Birkhäuser, 1996. https://doi.org/10.1007/978-1-4612-4146-1

[24] Y. Du, J. Su, C. Wang, The quasilinear Schrödinger-Poisson system, J. Math. Phys., 64 (2023), 071502. https://doi.org/10.1063/5.0150174

[25] X. Luo, Q. Wang, Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R³ , Nonlinear Anal. Real World Appl., 33 (2017), 19-32. https://doi.org/10.1016/j.nonrwa.2016.06.001

[26] K. Chang, Methods in Nonlinear Analysis , Springer, 2005. https://doi.org/10.1007/3-540-29232-2

[27] 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

[28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators , North-Holland, 1978. https://doi.org/10.1090/S0273-0979-1980-14753-9

[29] T. Bartsch, S. De Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math., 100 (2013), 75-83. https://doi.org/10.1007/s00013-012-0468-x

[30] M. Willem, Functional Analysis: Fundamentals and Applications , 2^nd edition, Springer, 2022. https://doi.org/10.1007/978-3-031-09149-0