This paper explores the effectiveness of personalized nudge strategies tailored to individual preferences, leveraging large-scale language models (LLMs). Specifically, we use LLMs to design personalized decoy-based nudges tailored to individual profiles and cultural contexts to encourage carbon offsets among air travelers. We evaluate their effectiveness through a large-scale survey experiment ($n=3,495$) in five countries (Germany, Singapore, the United States, China, and India). The LLM-based personalized nudges are more effective than a uniform setting, increasing carbon offset rates by 3-7% in Germany, Singapore, and the United States.