In this paper, a joint user scheduling and resource optimization framework for mobile edge computing (MEC) networks with asynchronous computing is studied. In the considered model, edge devices (EDs) offload their computational tasks to a MEC server utilizing energy harvested from the server. In particular, task offloading by EDs obey a particular order. Meanwhile, an asynchronous computing approach is adopted where the server executes a task once it is received. Intuitively, EDs offloading later can harvest more energy for transmission, while EDs uploading earlier have more computing time. An optimization problem of joint user scheduling, communication and computation resource management is formulated aiming to minimize the energy consumption of the server under the delay constraint. First, the general properties of computation resource allocation are investigated and a low-complexity frequency allocation algorithm is proposed. Then, through the General Benders Decomposition method, this mixed-integer non-linear programming problem is addressed by iteratively solving a primal problem related to continuous variables and a master problem associated with integer variables. Simulation results verify that the proposed algorithm yields significant performance gains compared with benchmark schemes, especially in resource-scarce scenarios.