Domain-specific Named Entity Recognition (NER), whose goal is to recognize domain-specific entities and their categories, provides an important support for constructing domain knowledge graphs. Currently, deep learning-based methods are widely used and effective in NER tasks, but due to the reliance on large-scale labeled data. As a result, the scarcity of labeled data in a specific domain will limit its application.Therefore, many researches started to introduce few-shot methods and achieved some results. However, the entity structures in specific domains are often complex, and the current few-shot methods are difficult to adapt to NER tasks with complex features.Taking the Chinese coal chemical industry domain as an example,there exists a complex structure of multiple entities sharing a single entity, as well as multiple relationships for the same pair of entities, which affects the NER task under the sample less condition.In this paper, we propose a Large Language Models (LLMs)-based entity recognition framework LLM-DER for the domain-specific entity recognition problem in Chinese, which enriches the entity information by generating a list of relationships containing entity types through LLMs, and designing a plausibility and consistency evaluation method to remove misrecognized entities, which can effectively solve the complex structural entity recognition problem in a specific domain.The experimental results of this paper on the Resume dataset and the self-constructed coal chemical dataset Coal show that LLM-DER performs outstandingly in domain-specific entity recognition, not only outperforming the existing GPT-3.5-turbo baseline, but also exceeding the fully-supervised baseline, verifying its effectiveness in entity recognition.