In this paper, we are first concerned with the exact number and expressions of the positive normalized solutions for the nonlinear Kirchhoff equation in which the frequency is unknown and to be determined by the solution. With these results, we further answer an open problem concerning the exact number of positive solutions to the Kirchhoff equation with fixed frequency. It is interesting that we observe some new phenomena, which are completely different from those for the corresponding nonlinear Schrödinger equation and which reveal the special influence of the nonlocal term.


  1. Kirchhoff equation
  2. normalized solution
  3. exact number of solutions

MSC codes

  1. 35J60 (35B09
  2. 35A01)

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 5424 - 5446
ISSN (online): 1095-7154


Submitted: 13 September 2021
Accepted: 3 May 2022
Published online: 22 September 2022


  1. Kirchhoff equation
  2. normalized solution
  3. exact number of solutions

MSC codes

  1. 35J60 (35B09
  2. 35A01)



Funding Information

Postdoctoral Research Foundation of China https://doi.org/10.13039/501100010031 : 2020M670272
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11771234, 11371212, 11025106

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