The core is a central solution concept in cooperative game theory, and therefore it is important to know under what conditions the core of a game is guaranteed to be non-empty. Two notions that prove to be very useful in this context are Linear Programming (LP) duality and convexity. In this work, we apply these tools to identify games with overlapping coalitions (OCF games) that admit stable outcomes. We focus on three notions of the core defined in (Chalkiadakis et al. 2010) for such games, namely, the conservative core, the refined core and the optimistic core. First, we show that the conservative core of an OCF game is non-empty if and only if the core of a related classic coalitional game is non-empty. This enables us to improve the result of (Chalkiadakis et al. 2010) by giving a strictly weaker sufficient condition for the non-emptiness of the conservative core. We then use LP duality to characterize OCF games with non-empty refined core; as a corollary, we show that the refined core of a game is non-empty as long as the superadditive cover of its characteristic function is convex. Finally, we identify a large class of OCF games that can be shown to have a non-empty optimistic core using an LP based argument.