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 $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ with no-flux boundary conditions. The source function $f$ in this system is a smooth function $f$ that satisfies $f(u)\le a-b{u}^{\eta }$ for $u\ge 0$. It is proven that $\eta \ge 1$ 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).