Electron. J. Differential Equations, Vol. 2019 (2019), No. 73, pp. 1-12.

Travelling solitary waves for boson stars

Guoqing Zhang, Ningning Song

Abstract:
In this article, we study the pseudo-relativistic Hartree equation
$$ i\partial_{t}\psi=(\sqrt{-\Delta+m^2}-m)\psi -(\frac{e^{-\mu|x|}}{4\pi|x|}\ast|\psi|^2)\psi,\quad \text{on }\mathbb{R}^3, $$
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].

Submitted July 3, 2018. Published May 28, 2019.
Math Subject Classifications: 35Q40, 35Q55, 47J35.
Key Words: Travelling solitary wave; boson star equation; critical stellar mass.

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Guoqing Zhang
College of Sciences
University of Shanghai for Science and Technology
Shanghai 200093, China
email: shzhangguoqing@126.com
Ningning Song
College of Sciences
University of Shanghai for Science and Technology
Shanghai 200093, China
email: 787661389@qq.com

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