The fifth generation (5G) of cellular networks provides the enabling environment for the Internet of Things (IoT) applications. Hence, the vast proliferation of 5G-enabled IoT devices and services led to an overwhelming growth of data traffic that could saturate the core network’s backhaul links. Nowadays, caching is an unavoidable technique to solve this issue, whereby popular contents are stored on edge nodes near to end-users. There exist several initiatives to motivate caching actors for improving the caching process, but not designed for the real-world competitive caching market. In this work, we propose an incentive caching strategy in a 5G-enabled IoT network by considering a completely competitive caching scenario with multiple 5G mobile network operators (MNOs) and multiple content providers (CPs). The MNOs manage a set of edge caches on their base stations and they are competing to fill these caching resources, while the CPs detain a set of popular contents and are in conflict to rent the MNOs’ caches. Each MNO aims to maximize its monetary profit and offload its backhaul links, as each CP aims to improve the quality of experience (QoE) of its end-users. Then, we formulate a multi-leader multi-follower Stackelberg game to model the interaction between MNOs and CPs and define the different players’ utilities. Subsequently, we propose an iterative algorithm based on the convex optimization method to investigate the Stackelberg equilibrium. Finally, the numerical results of the different experimentations demonstrate that our game-based incentive strategy can significantly alleviate the backhaul links while improving the user QoE.