The Lightning Network is the most widely used payment channel network (PCN) to date, making it an attractive attack surface for adversaries. In this paper, we analyze the Lightning Network's PCN topology and investigate its resilience towards random failures and targeted attacks. In particular, we introduce the notions of channel exhaustion and node isolation attacks and show that the Lightning Network is susceptible to these attacks. In a preliminary analysis, we confirm that the Lightning Network can be classified as a small-world and scale-free network. Based on these findings, we develop a series of strategies for targeted attacks and introduce metrics that allow us to quantify the adversary's advantage. Our results indicate that an attacker who is able to remove a certain number of nodes should follow a centrality-based strategy, while a resource-limited attacker who aims for high efficiency should employ a highest ranked minimum cut strategy.