We propose a Green Cloudlet Network (GCN) architecture to provide seamless Mobile Cloud Computing (MCC) services to User Equipments (UEs) with low latency in which each cloudlet is powered by both green and brown energy. Fully utilizing green energy can significantly reduce the operational cost of cloudlet providers. However, owing to the spatial dynamics of energy demand and green energy generation, the energy gap among different cloudlets in the network is unbalanced, i.e., some cloudlets' energy demands can be fully provided by green energy but others need to utilize on-grid energy (i.e., brown energy) to satisfy their energy demands. We propose a Green-energy awarE Avatar migRation (GEAR) strategy to minimize the on-grid energy consumption in GCN by redistributing the energy demands via Avatar migration among cloudlets according to cloudlets' green energy generation. Furthermore, GEAR ensures the Service Level Agreement (SLA) in terms of the maximum Avatar propagation delay by avoiding Avatars hosted in the remote cloudlets. We formulate the GEAR strategy as a mixed integer linear programming problem, which is NP-hard, and thus apply the Branch and Bound search to find its sub-optimal solution. Simulation results demonstrate that GEAR can save on-grid energy consumption significantly as compared to the Follow me AvataR (FAR) migration strategy, which aims to minimize the propagation delay between an UE and its Avatar.