The recent advance in radio-frequency (RF) wireless energy transfer (WET) has motivated the study of wireless powered communication network (WPCN), in which distributed wireless devices are powered via dedicated WET by the hybrid access-point (H-AP) in the downlink (DL) for uplink (UL) wireless information transmission (WIT). In this paper, by exploiting the cognitive radio (CR) technique, we study a new type of CR enabled secondary WPCN, called cognitive WPCN, under spectrum sharing with the primary wireless communication system. In particular, we consider a cognitive WPCN, consisting of one single H-AP with constant power supply and distributed users, shares the same spectrum for its DL WET and UL WIT with an existing primary communication link, where the WPCN's WET/WIT and the primary link's WIT may interfere with each other. Under this new setup, we propose two coexisting models for spectrum sharing of the two systems, namely underlay and overlay based cognitive WPCNs, depending on different types of knowledge on the primary user transmission available at the cognitive WPCN. For each model, we maximize the sum-throughput of the cognitive WPCN by optimizing its transmission under different constraints applied to protect the primary user transmission. Analysis and simulation results are provided to compare the sum-throughput of the cognitive WPCN versus the achievable rate of the primary user in two coexisting models. It is shown that the overlay based cognitive WPCN outperforms the underlay based counterpart, thanks to its fully cooperative WET/WIT design with the primary WIT, while it also requires higher complexity for implementation.