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Nowadays a wide range of applications is constrained by low-latency requirements that cloud infrastructures cannot meet. Multi-access Edge Computing (MEC) has been proposed as the reference architecture for executing applications closer to users and reduce latency, but new challenges arise: edge nodes are resource-constrained, the workload can vary significantly since users are nomadic, and task complexity is increasing (e.g., machine learning inference). To overcome these problems, the paper presents NEPTUNE, a serverless-based framework for managing complex MEC solutions. NEPTUNE i) places functions on edge nodes according to user locations, ii) avoids the saturation of single nodes, iii) exploits GPUs when available, and iv) allocates resources (CPU cores) dynamically to meet foreseen execution times. A prototype, built on top of K3S, was used to evaluate NEPTUNE on a set of experiments that demonstrate a significant reduction in terms of response time, network overhead, and resource consumption compared to three state-of-the-art approaches.