The emergence of generative artificial intelligence (GenAI) marks a significant breakthrough in the realm of AI. Recently, GenAI and large language models (LLMs) have garnered tremendous attention due to their capability to automatically generate data based on the given original patterns and dataset. However, traditional GenAI-LLMs mechanisms may result in low-quality output content and considerable creation time. The Internet of agents (IoA) can address these challenges by providing a flexible and scalable platform for integrating diverse agents, enabling seamless communication and coordination in mobile environments. Therefore, in this work, we explore the integration of multi-agent GenAI-LLMs enlightened by IoA. Specifically, we first provide a brief introduction to multi-agent GenAI-LLMs and their applications in different domains. Then, we demonstrate the potential of deploying the multi-agent GenAI-LLMs in mobile edge networks. Subsequently, we discuss the emerging applications and challenges when deploying multi-agent GenAI-LLMs in mobile edge networks. In the following, we propose a novel multi-agent GenAI-LLM architecture for mobile edge networks. Moreover, we conduct a case study to show the effectiveness of the proposed architecture by applying it to generate high-quality solutions in unmanned aerial vehicle (UAV) networks. Finally, several potential research directions for GenAI-LLMs in mobile edge networks are discussed.