With the popularity of online shopping, an increasing number of buyers and sellers are basing their decisions on mass online reviews. As an efficient approach to obtain fine-grained sentiment information, cross-domain aspect-based sentiment classification has received extensive attention from academia and industry. However, this task encounters challenges such as limited labeled data or extreme differences in sentiment distributions between source and target domains. To address these issues, we propose a novel approach named CSC-PLDAT (Cross-domain Aspect-based Sentiment Classification with Prompt Learning and Domain Adversarial Training). Specifically, CSC-PLDAT incorporates a hybrid prompt comprising transferable and task-specific components. Based on this prompt, our model can effectively transfer domain-specific sentiment information. Meanwhile, it can adapt to the situation with limited labeled or skewed sentiment distributions. Additionally, considering various sentiment words can express identical sentiment polarities, we employ domain adversarial training to generate domain-independent sentiment representations for sentiment classification. The experiment is conducted on four popular benchmark datasets. The results demonstrate that our model outperforms the state-of-the-art methods, with an average micro-F1 score of 71.45% in cross-domain aspect-based sentiment classification. Furthermore, our model exhibits promising outcomes, particularly in scenarios characterized by limited labeled data and skewed sentiment distributions.