Global boundedness of solutions to the chemotaxis system with general sensitivity and nonlinear production

In this paper, we consider the following chemotaxis system \begin{equation*} \left\{ \begin{split} &n_{t}=\nabla\cdot((n+1)^{m-1}\nabla n)-\nabla\cdot\left(\frac{n\nabla v}{(1+n)^\alpha}\right),&&x\in \Omega ,t> 0,\\ &0=\Delta v-v+n^\gamma,&&x\in \Omega ,t> 0\\ \end{split} \right. \end{equation*} in a bounded domain $\Omega\subset\mathbb{R}^N, N\geq2$ with smooth boundary. It is shown that for all reasonably regular initial data, if $\alpha>max\left\{0,\gamma-m-\frac{2}{N}+1\right\}$, $\gamma>0$ and $m\geq1$, then the problem possesses a unique global bounded classical solution.