Exact Number of Positive Solutions for the Kirchhoff Equation

Normalized solutions for the Kirchhoff equation on noncompact metric graphs *

“…In the present work, we consider the normalized solutions for the Kirchhoff equation on noncompact metric graphs. We first study the problem on the half line and the real line, and obtain the exact number and expressions of positive normalized solutions (We can see [42] for more general setting in R N ). Then we consider the normalized ground states of (1.1) on more general noncompact metric graphs within the variational frame.…”

“…In this section, we are devoted to the study of positive solutions for (1.1) on the real line and half line. We can see [42] for more discussion in R N . Hence we only give the outline for completeness.…”

Normalized Solutions to Kirchhoff Equation with Nonnegative Potential

“…After that, for 2 < p < 2 + 4 N , Zeng and Zhang [32] proved the existence and uniqueness of the minimizer to (1.7) for any c > 0, while for 2 + 4 N < p < 2 + 8 N the authors proved that there exists a threshold mass c * > 0 such that for any c ∈ (0, c * ) there is no minimizer and for c > c * there is a unique minimizer. Moreover, a precise formula for the minimizer and the threshold value c * is given according to the mass c. In a recent paper [24], Qi and Zou first obtained the exact number and expressions of the positive normalized solutions to (K ∞,c ) for 2 < p ≤ 2 * , and then answered an open problem about the exact number of positive solutions of the Kirchhoff equation with fixed frequency. In particular, with trapping potential V and p = 2 + 8 N , Hu and Tang [13] considered the concentration behavior and local uniqueness of the normalized solution for mass critical Kirchhoff equations as a tending to 0.…”