On ground states for the L^2-critical boson star equation

TL;DR

This paper studies ground state solutions of the L^2-critical boson star equation, proving their analyticity and radial symmetry, and discusses related uniqueness and nondegeneracy results in the context of fractional Laplacian equations.

Contribution

It establishes analyticity and radial symmetry of ground states for the L^2-critical boson star equation in three dimensions.

Findings

Ground states are analytic and radially symmetric.

Previous claims of uniqueness and nondegeneracy are addressed with references to recent work.

The paper discusses the gap in earlier proofs and points to a broader context of fractional Laplacian equations.

Abstract

We consider ground state solutions $u \geq 0$ for the $L^2$-critical boson star equation $$ \sqrt{-\Delta} \, u - \big (|x|^{-1} \ast |u|^2 \big) u = -u \quad {in $\R^3$}. $$ We prove analyticity and radial symmetry of $u$. In a previous version of this paper, we also stated uniqueness and nondegeneracy of ground states for the $L^2$-critical boson star equation in $\R^3$, but the arguments given there contained a gap. However, we refer to our recent preprint \cite{FraLe} in {\tt arXiv:1009.4042}, where we prove a general uniqueness and nondegeneracy result for ground states of nonlinear equations with fractional Laplacians in $d=1$ space dimension.