Large Language Models (LLMs) have demonstrated an impressive level of general knowledge. However, they often struggle in highly specialized and sensitive domains such as drug discovery and rare disease research due to the lack of expert knowledge, which is often costly to obtain. In this paper, we propose a novel framework (PU-ADKA) designed to efficiently enhance domain-specific LLMs by actively engaging domain experts within a fixed budget. Unlike traditional fine-tuning approaches, PU-ADKA proactively identifies and queries the most appropriate expert from a team, taking into account each expert's availability, competency, knowledge boundaries, and consultation cost. We train PU-ADKA using simulations on PubMed publication data and validate it through domain expert interactions, showing promising improvements in LLM domain knowledge acquisition. Furthermore, our experiments with a real-world drug development team validate that PU-ADKA can significantly enhance LLM performance in specialized domains while adhering to strict budget constraints. In addition to outlining our methodological innovations and experimental results, we release a new benchmark dataset, CKAD, for cost-effective LLM domain knowledge acquisition to foster further research in this challenging area.