Asymptotic Behaviors of Chandrasekhar Variational Problem for Neutron Stars With Slater-Type Modification

Abstract

In this paper, we consider the Chandrasekhar variational model for neutron stars with defocusing Slater-type modifications. First, we show the existence and nonexistence of the ground state ${\rho }_{\epsilon }$ by concentration–compactness method, and particularly use two auxiliary functions to prove the strongly binding inequality. Second, we characterize perturbation limit behaviors of ground states ${\rho }_{\epsilon }$ as $\epsilon \to 0^{+}$ and obtain two different blow-up profiles corresponding to two limit equations for $N=N_{\ast }$ and $N>N_{\ast }$, where $\epsilon$ is a parameter corresponding to Slater-type modifications, and $N_{\ast }$ is a threshold value related to the Chandrasekhar limit. Finally, we study the limit behaviors for $N\ge N_{\ast }$ as $\epsilon \to +\infty$, using some iterate arguments, we obtain a vanishing rate for ${\rho }_{\epsilon }$ that $\parallel {\rho }_{\epsilon }{\parallel }_{L^{\infty }}\lesssim {\epsilon }^{-\frac{1}{\alpha -1}}$ as $\epsilon \to +\infty$ for any . Moreover, we characterize the limit behaviors of the energy $E_{\epsilon }\left(N\right)$ with respect to $\epsilon$, and show that ${\lim }_{\epsilon \to +\infty }{E}_{\epsilon }\left(N\right)=mN$, $E_{\epsilon }\left(N\right)$ is concave and strictly monotonically increasing with respect to $\epsilon >0$ in some case.