Existing industrial and household robots encounter challenges in adapting to intricate scenarios and manipulating diverse objects, leading to insufficient adaptability. Furthermore, current research in grasp detection is limited to simplified background environments, hindering the ability to meet the demands of complex real-world applications. To address these challenges, we propose a multi-object background-adaptive Grasp detection network named MD-Grasp, which is based on Mamba & attention fusion block and Dense block. MD-Grasp integrates Dense Block for the extraction of local features, effectively accentuating target objects situated in the foreground. On the other hand, the Mamba & Attention Fusion Block enhances global feature modeling, thereby highlighting critical regions within the foreground. Extensive experiments demonstrate that the proposed MD-Grasp exhibits significant advantages when confronted with complex and variable objects. In single-object grasping, the MD-Grasp scheme achieved accuracy rates of 98.7% and 97.7% on the Cornell dataset, and 95.5% on the Jacquard dataset. In multi-object grasping, the MD-Grasp scheme improved the average accuracy rate by 1% to 3% on the CBRGD dataset compared to the baseline. Experiments conducted in real household environments validated that MD-Grasp can maintain high grasping accuracy.