Boson Stars with Long-range Perturbations

Abstract

We consider the Boson star equation with long-range perturbation given by $$i{\partial }_t\psi =\sqrt{-△+m^2}\psi +\beta (\frac{1}{|x{|}^{\alpha }}\ast |\psi {|}^2)\psi -(\frac{1}{|x|}\ast |\psi {|}^2)\psi \text{on R3,}$$ 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 $\Vert {\psi }_0{\Vert }_2^2=N_c$, 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 $\beta \to 0^{+}$.

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