| Summary: | This paper deals with a parabolic-elliptic chemotaxis-growth system with nonlinear sensitivity
\begin{equation*}\label{1a}
\begin{cases}
u_t=\Delta u-\chi\nabla\cdot(\psi(u)\nabla v)+f(u), &(x,t)\in \Omega\times (0,\infty), \\
0=\Delta v-v+g(u), &(x,t)\in \Omega\times (0,\infty),
\end{cases}
\end{equation*}
under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi>0$, the chemotactic sensitivity $\psi(u)\leq(u+1)^{q}$ with $q>0$, $g(u)\leq(u+1)^{l}$ with $l\in \mathbb{R}$ and $f(u)$ is a logistic source. The main goal of this paper is to extend a previous result on global boundedness by Zheng et al. [J. Math. Anal. Appl. 424(2015), 509–522] under the condition that $1\leq q+l<\frac{2}{n}+1$ to the case $q+l<1$.
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