Motivated by the recent efforts in extending long term evolution (LTE) to the unlicensed spectrum, we propose a novel spectrum sharing framework for the coopetition (i.e., cooperation and competition) between LTE and Wi-Fi in the unlicensed band. Basically, the LTE network can choose to work in one of the two modes: in the competition mode, it randomly accesses an unlicensed channel, and interferes with the Wi-Fi access point using the same channel; in the cooperation mode, it onloads the Wi-Fi users' traffic in exchange for the exclusive access of the corresponding channel. We design a second-price reverse auction mechanism, which enables the LTE provider and the Wi-Fi access point owners (APOs) to effectively negotiate the operation mode. Specifically, the LTE provider is the auctioneer (buyer), and the APOs are the bidders (sellers) who compete to sell the rights of onloading the APOs' traffic to the LTE provider. In Stage I of the auction, the LTE provider announces a reserve rate, which is the maximum data rate that it is willing to allocate to the APOs in the cooperation mode. In Stage II of the auction, the APOs submit their bids, which indicate the data rates that they would like the LTE provider to offer in the cooperation mode. We show that the auction involves allocative externalities, i.e., the cooperation between the LTE provider and one APO benefits other APOs who are not directly involved in this cooperation. We characterize the APOs' unique equilibrium bidding strategies in Stage II, and analyze the LTE provider's optimal reserve rate in Stage I. Numerical results show that our framework improves the payoffs of both the LTE provider and the APOs comparing with a benchmark scheme. In particular, our framework increases the LTE provider's payoff by 70% on average, when the LTE provider has a large throughput and a small data rate discounting factor. Moreover, our framework leads to a close-to-optimal social welfare under a large LTE throughput.