# The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars

## TL;DR

This paper proves that for small enough stellar mass, the ground states of pseudo-relativistic Boson stars are unique, confirming a longstanding conjecture by Lieb and Yau in a specific case.

## Contribution

It establishes the validity of the Lieb-Yau conjecture on the uniqueness of ground states for small stellar masses in pseudo-relativistic Boson stars.

## Key findings

- Ground states exist if and only if stellar mass N is positive and less than a critical value N*.
- The Lieb-Yau conjecture on uniqueness is confirmed for sufficiently small N.
- The proof applies to the specific case where N is small enough.

## Abstract

It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass $N>0$ satisfies $N<N^*$, where the finite constant $N^*$ is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each $N<N^*$. In this paper, we prove that the above uniqueness conjecture holds for the particular case where $N>0$ is small enough.