To improve the network switching performance and efficiency of mobile phone terminals and establish an efficient mobile communication network connection, this paper constructs the SDN+MPTCP+CP (Software Defined Network, Multi-Path TCP, and Mobile Terminal) mobile communication network model and designs a network switching algorithm with a preselected available access point name (APN) based on the potential game method. The constructed network model integrates a 5G mobile communication network, satellite communication, the SDN network, and the MPTCP multi-way communication technology. APN access point is preselected by using Kalman filtering theory, and dual problems are resolved with the Lagrange function. To determine the MPTCP sub-flow transmission path, the differential derivative calculation is introduced. The performance of the network switching strategy is evaluated based on the Jacobian matrix. Then, the game coefficient is designed, and the game function is calculated. A potential game balance point is found, and an updating strategy is formulated to determine the best APN access point. The simulation network model is constructed, and the parameters of the performance evaluation are defined to find the performance comprehensively. The experimental results demonstrated the extreme reliability, stability, and compatibility of the proposed algorithm.