Stable solitary waves for pseudo-relativistic Hartree equations with short range potential

Abstract

We consider solitary waves with prescribed stellar mass for the pseudo-relativistic Hartree equation, which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. This leads to the study of the following nonlocal elliptic equation −Δ+m2u−(|x|−γ∗|u|2)u=μu,x∈R3under the normalized constraint ∫R3|ψ(t,x)|2dx=N,where γ∈(1,2), m>0, N>0 denotes the stellar mass of Boson stars and the frequency μ∈R is unknown and appears as Lagrange multiplier. In contrast to the mass-subcritical case γ∈(0,1) and mass-critical case γ=1, the corresponding pseudo-relativistic Hartree energy functional is always unbounded on the L2 sphere for any N>0, we prove that the above problem admits at least one solution uN, which is a ground state if N or m is sufficiently small. Furthermore, the stability of the corresponding solitary wave for the related time-dependent pseudo-relativistic Hartree equation is given. In addition, we also give an accurate description of the limiting behavior of uN as N→0+ and m→0+, respectively. The main contribution of this paper is to extend the main results in Guo and Zeng (2017), Lenzmann (2009) and Coti Zelati and Nolasco (2013) from long range potential case γ∈(0,1] to short range potential case γ∈(1,2).