Content Delivery Network (CDN) solutions have been progressively integrated to many network infrastructures. However, they show technological limitations when dealing with an increasing number of bandwidth-hungry and demanding applications. One of the most effective ways to improve the performance of a CDN is to efficiently manage its traffic by implementing relevant request routing solutions. To this end, we present in this paper a flexible scheme and an optimization algorithm, based on Lyapunov theory, for request routing in CDNs. Our online approach provides a stable quality of service to clients, while improving content delivery delays. It also reduces data transport costs for operators and outperforms existing techniques in terms of peak traffic management.