The rapidly increasing demand in indoor small cell networks has given rise to the concept of local 5G operator (OP) for local service delivery. In this regard, we develop a novel game-theoretic framework with geometric programming to model and analyze cache-enabled small cell base stations (SBSs) with infrastructure sharing for local 5G OP networks. In such a network, the local 5G OP provides wireless network in indoor area and rent out the infrastructure which are RAN and cache storage to multiple mobile network operators (MNOs) while guarantee the quality-of-experience (QoE) at the users (UEs) of MNOs. We formulate a Stackelberg game model where the local 5G OP is the leader and the MNOs are the followers. The local 5G OP aims to maximize its profit by optimizing its infrastructure rental fee, and the MNOs aim to minimize their renting cost of infrastructure by minimizing the "cache intensity" subject to latency constraint at each UE. Here, the cache intensity is defined as the product of the number of SBSs per unit area and the number of popular files stored in each SBS. The optimization problems of the local 5G OP and the MNOs are transformed into geometric programming. Accordingly, the subgame perfect equilibrium of Stackelberg game is obtained through the succesive geometric programming (SGP) method. Since the MNOs share their rented infrastructure, for cost sharing, we apply the concept of Shapley value to divide the cost among the MNOs. We show that the cost sharing problem can be mapped into a simplified "airport runway cost sharing problem", in which the Shapley value can be computed efficiently. Finally, we present an extensive performance evaluation that reveals interesting insights into designing resource sharing with edge caching in local 5G OP networks.