Travelling solitary waves for boson stars

Abstract

In this article, we study the pseudo-relativistic Hartree equation i∂tψ = (√-∆ + m2 - m)ψ - (e-μ|x| / 4π|x| ∗ |ψ|2)ψ, on ℝ3, which describes the dynamics of pseudo-relativistic boson stars with rest mass m > 0 in the mean-field limit. Based on Ekeland variational principle, concentration-compactness lemma and Gagliardo-Nirenberg inequality, we prove existence of travelling solitary waves under the critical stellar mass. In addition to their existence, we obtain orbital stability by using a general idea presented in Cazenave and Lions [2].