  For a class of Kirchhoff functional, we first give a complete classification
with respect to the exponent $p$ for its $L^2$-normalized critical points, and
show that the minimizer of the functional, if exists, is unique up to
translations. Secondly, we search for the mountain pass type critical point for
the functional on the $L^2$-normalized manifold, and also prove that this type
critical point is unique up to translations. Our proof relies only on some
simple energy estimates and avoids using the concentration-compactness
principles. These conclusions extend some known results in previous papers.
