Boson Stars with Long-range Perturbations

We consider the Boson star equation with long-range perturbation given by $$i\partial_t \psi=\sqrt{-\triangle+m^2}\,\psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$,}$$ where $\frac{1}{|x|^\alpha} (0<\alpha<1)$ denotes the long-range potential. In contrast to the well known fact that for $\beta=0$ no maximal ground state solitary wave exists when the partical number $N=N_c$ (Chandrasekhar limiting mass) [E.H. Lieb, H.T. Yau, \emph{Commun. Math. Phys.}, 112 (1987), pp: 147-174 ], we show that for $\beta>0$ and small enough, there exists at least one maximal ground state at $N=N_c$. Moreover, for $\beta>0$, we find that for initial value $\|\psi_0\|^2_2=N_c$, the solution $\psi(t)$ is global well-posedness, and we obtain an "orbital stability" of those maximal ground state solitary waves in some sense, which implies that such long-range perturbation pushes the Boson star system more stable. Finally, we analyse blow-up behaviours of maximal ground states when $\beta\rightarrow 0^+$.