A further study on an attraction-repulsion chemotaxis system with logistic source

This paper is concerned with the attraction-repulsion chemotaxis system (1.1) define on a bounded domain $ \Omega \subset \mathbb{R}^N(N\geq 1) $ with no-flux boundary conditions. The source function $ f $ in this system is a smooth function $ f $ that satisfies $ f(u)\leq a-bu^\eta $ for $ u\geq0 $. It is proven that $ \eta\geq1 $ is sufficient to ensure the boundedness of the solution when $ r < \frac{4(N+1)}{N(N+2)} $ is in the balance case $ \chi\alpha = \xi\gamma $, which improve the relevant results presented in papers such as Li and Xiang (2016), Xu and Zheng (2018), Xie and Zheng (2021), and Tang, Zheng and Li (2023).