Mobile edge computing has emerged as a promising paradigm to complement the computing and energy resources of mobile devices. In this computing paradigm, mobile devices offload their computing tasks to nearby edge servers, which can potentially reduce their energy consumption and task completion delay. In exchange for processing the computing tasks, edge servers expect to receive a payment that covers their operating costs and allows them to make a profit. Unfortunately, existing works either ignore the payments to the edge servers, or ignore the task processing delay and energy consumption of the mobile devices. To bridge this gap, we propose an auction to allocate edge servers to mobile devices that is executed by a pair of deep neural networks. Our proposed auction maximizes the profit of the edge servers, and satisfies the task processing delay and energy consumption constraints of the mobile devices. The proposed deep neural networks also guarantee that the mobile devices are unable to unfairly affect the results of the auctions. Our extensive simulations show that our proposed auction mechanism increases the profit of the edge servers by at least 50% compared to randomized auctions, and satisfies the task processing delay and energy consumption constraints of mobile devices.