# Existence and Uniqueness of Normalized Solutions for the Kirchhoff equation

## TL;DR

This paper investigates the existence and uniqueness of normalized solutions for the Kirchhoff equation, classifying critical points based on the exponent and establishing uniqueness of minimizers and mountain pass solutions without concentration-compactness.

## Contribution

It provides a complete classification of $L^2$-normalized critical points for Kirchhoff functionals and proves the uniqueness of minimizers and mountain pass solutions up to translations, extending previous results.

## Key findings

- Classification of critical points based on exponent p
- Uniqueness of minimizers up to translations
- Uniqueness of mountain pass critical points up to translations

## Abstract

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.