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Mobile Edge Computing (MEC) has emerged as a promising supporting architecture providing a variety of resources to the network edge, thus acting as an enabler for edge intelligence services empowering massive mobile and Internet of Things (IoT) devices with AI capability. With the assistance of edge servers, user equipments (UEs) are able to run deep neural network (DNN) based AI applications, which are generally resource-hungry and compute-intensive, such that an individual UE can hardly afford by itself in real time. However the resources in each individual edge server are typically limited. Therefore, any resource optimization involving edge servers is by nature a resource-constrained optimization problem and needs to be tackled in such realistic context. Motivated by this observation, we investigate the optimization problem of DNN partitioning (an emerging DNN offloading scheme) in a realistic multi-user resource-constrained condition that rarely considered in previous works. Despite the extremely large solution space, we reveal several properties of this specific optimization problem of joint multi-UE DNN partitioning and computational resource allocation. We propose an algorithm called Iterative Alternating Optimization (IAO) that can achieve the optimal solution in polynomial time. In addition, we present rigorous theoretic analysis of our algorithm in terms of time complexity and performance under realistic estimation error. Moreover, we build a prototype that implements our framework and conduct extensive experiments using realistic DNN models, whose results demonstrate its effectiveness and efficiency.