Over the past decade, community detection in overlapping un-weighted networks, where nodes can belong to multiple communities, has been one of the most popular topics in modern network science. However, community detection in overlapping weighted networks, where edge weights can be any real value, remains challenging. In this article, we propose a generative model called the weighted degree-corrected mixed membership (WDCMM) model to model such weighted networks. This model adopts the same factorization for the expectation of the adjacency matrix as the previous degree-corrected mixed membership (DCMM) model. Our WDCMM extends the DCMM from un-weighted networks to weighted networks by allowing the elements of the adjacency matrix to be generated from distributions beyond Bernoulli. We first address the community membership estimation of the model by applying a spectral algorithm and establishing a theoretical guarantee of consistency. Then, we propose overlapping weighted modularity to measure the quality of overlapping community detection for both assortative and dis-assortative weighted networks. To determine the number of communities, we incorporate the algorithm into the proposed modularity. We demonstrate the advantages of the model and the modularity through applications to simulated data and real-world networks.