Drug repositioning identifies novel therapeutic potentials for existing drugs and is considered an attractive approach due to the opportunity for reduced development timelines and overall costs. Prior computational methods usually learned a drug's representation from an entire graph of drug-disease associations. Therefore, the representation of learned drugs representation are static and agnostic to various diseases. However, for different diseases, a drug's mechanism of actions (MoAs) are different. The relevant context information should be differentiated for the same drug to target different diseases. Computational methods are thus required to learn different representations corresponding to different drug-disease associations for the given drug. In view of this, we propose an end-to-end partner-specific drug repositioning approach based on graph convolutional network, named PSGCN. PSGCN firstly extracts specific context information around drug-disease pairs from an entire graph of drug-disease associations. Then, it implements a graph convolutional network on the extracted graph to learn partner-specific graph representation. As the different layers of graph convolutional network contribute differently to the representation of the partner-specific graph, we design a layer self-attention mechanism to capture multi-scale layer information. Finally, PSGCN utilizes sortpool strategy to obtain the partner-specific graph embedding and formulates a drug-disease association prediction as a graph classification task. A fully-connected module is established to classify the partner-specific graph representations. The experiments on three benchmark datasets prove that the representation learning of partner-specific graph can lead to superior performances over state-of-the-art methods. In particular, case studies on small cell lung cancer and breast carcinoma confirmed that PSGCN is able to retrieve more actual drug-disease associations in the top prediction results. Moreover, in comparison with other static approaches, PSGCN can partly distinguish the different disease context information for the given drug.