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E-commerce platforms provide entrances for customers to enter mini-apps that can meet their specific shopping requirements. Trigger items displayed on entrance icons can attract more entering. However, conventional Click-Through-Rate (CTR) prediction models, which ignore user instant interest in trigger item, fail to be applied to the new recommendation scenario dubbed Trigger-Induced Recommendation in Mini-Apps (TIRA). Moreover, due to the high stickiness of customers to mini-apps, we argue that existing trigger-based methods that over-emphasize the importance of trigger items, are undesired for TIRA, since a large portion of customer entries are because of their routine shopping habits instead of triggers. We identify that the key to TIRA is to extract customers' personalized entering intention and weigh the impact of triggers based on this intention. To achieve this goal, we convert CTR prediction for TIRA into a separate estimation form, and present Deep Intention-Aware Network (DIAN) with three key elements: 1) Intent Net that estimates user's entering intention, i.e., whether he/she is affected by the trigger or by the habits; 2) Trigger-Aware Net and 3) Trigger-Free Net that estimate CTRs given user's intention is to the trigger-item and the mini-app respectively. Following a joint learning way, DIAN can both accurately predict user intention and dynamically balance the results of trigger-free and trigger-based recommendations based on the estimated intention. Experiments show that DIAN advances state-of-the-art performance in a large real-world dataset, and brings a 9.39% lift of online Item Page View and 4.74% CTR for Juhuasuan, a famous mini-app of Taobao.