In biomedical fields, one named entity may consist of a series of non-adjacent tokens and overlap with other entities. Previous methods recognize discontinuous entities by connecting entity fragments or internal tokens, which face challenges of error propagation and decoding ambiguity due to the wide variety of span or word combinations. To address these issues, we deeply explore discontinuous entity structures and propose an effective Gap-aware grid tagging model for Discontinuous Named Entity Recognition, named GapDNER. Our GapDNER innovatively applies representation learning on the context gaps between entity fragments to resolve decoding ambiguity and enhance discontinuous NER performance. Specifically, we treat the context gap as an additional type of span and convert span classification into a token-pair grid tagging task. Subsequently, we design two interactive components to comprehensively model token-pair grid features from both intra- and inter-span perspectives. The intra-span regularity extraction module employs the biaffine mechanism along with linear attention to capture the internal regularity of each span, while the inter-span relation enhancement module utilizes criss-cross attention to obtain semantic relations among different spans. At the inference stage of entity decoding, we assign a directed edge to each entity fragment and context gap, then use the BFS algorithm to search for all valid paths from the head to tail of grids with entity tags. Experimental results on three datasets demonstrate that our GapDNER achieves new state-of-the-art performance on discontinuous NER and exhibits remarkable advantages in recognizing complex entity structures.