Nearest neighbor search is fundamental to a wide range of applications. Since the exact nearest neighbor search suffers from the "curse of dimensionality", approximate approaches, such as Locality-Sensitive Hashing (LSH), are widely used to trade a little query accuracy for a much higher query efficiency. In many scenarios, it is necessary to perform nearest neighbor search under multiple weighted distance functions in high-dimensional spaces. This paper considers the important problem of supporting efficient approximate nearest neighbor search for multiple weighted distance functions in high-dimensional spaces. To the best of our knowledge, prior work can only solve the problem for the $l_2$ distance. However, numerous studies have shown that the $l_p$ distance with $p\in(0,2)$ could be more effective than the $l_2$ distance in high-dimensional spaces. We propose a novel method, WLSH, to address the problem for the $l_p$ distance for $p\in(0,2]$. WLSH takes the LSH approach and can theoretically guarantee both the efficiency of processing queries and the accuracy of query results while minimizing the required total number of hash tables. We conduct extensive experiments on synthetic and real data sets, and the results show that WLSH achieves high performance in terms of query efficiency, query accuracy and space consumption.