Integrating terrestrial cellular network with Low Earth Orbit (LEO) satellite constellation has been a popular trend in beyond 5G and 6G eras, called Satellite-Terrestrial Integrated Core Network (STICN). Core Network (CN) is an essential component responsible for authentication, security, mobility, data routing, etc. However, the terrestrial CN is designed for infrastructure-fixed and user-moving scenarios, which would cause the STICN to experience signaling storms and service interruptions when users access by rapid satellites. In this paper, we propose a distributed lightweight stateless satellite CN architecture, which could fit with the dynamic and limited-resource instincts of LEO satellites. And it can cooperate with terrestrial CN to provide seamless services. Firstly, the contexts of Network Functions (NFs) are decoupled from themselves and are managed in a common repository. Moreover, we design a cooperation mechanism between NFs to avoid frequent transmission of context and service interruption. Finally, extensive experiments are carried out on semi-physical simulation environments. The STICN performance could be improved by selecting the optimal number and location of each NF. Our evaluation shows that the proposed scheme could reduce the delay of the handover procedure by 37% and is more resilient compared with terrestrial CN.