Exact Number of Positive Solutions for the Kirchhoff Equation

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.