This article considers the no-flux attraction-repulsion chemotaxis model {ut=∇⋅((u+1)m1−1∇u−χu(u+1)m2−2∇v+ξu(u+1)m3−2∇w),x∈Ω,t>0,0=Δv+f(u)−βv,x∈Ω,t>0,0=Δw+g(u)−δw,x∈Ω,t>0\documentclass[12pt]{minimal}
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\begin{document}$$ \left \{ \textstyle\begin{array}{l} \begin{aligned} &u_{t} = \nabla \cdot \big((u+1)^{m_{1}-1}\nabla u-\chi u(u+1)^{m_{2}-2} \nabla v+\xi u(u+1)^{m_{3}-2}\nabla w\big),& x\in \Omega ,\ t>0&, \\ & 0=\Delta v+f(u)-\beta v, & x\in \Omega ,\ t>0&, \\ & 0=\Delta w+g(u)-\delta w, & x\in \Omega ,\ t>0& \end{aligned} \end{array}\displaystyle \right . $$\end{document} defined in a smooth and bounded domain Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$\Omega \subset \mathbb{R}^{n}$\end{document} (n≥2\documentclass[12pt]{minimal}
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\begin{document}$n\ge 2$\end{document}) with m1,m2,m3∈R\documentclass[12pt]{minimal}
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\begin{document}$m_{1},m_{2},m_{3}\in \mathbb{R}$\end{document}, χ,ξ,β,δ>0\documentclass[12pt]{minimal}
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\begin{document}$\chi ,\xi ,\beta ,\delta >0$\end{document}. The functions f(u)\documentclass[12pt]{minimal}
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\begin{document}$f(u)$\end{document}, g(u)\documentclass[12pt]{minimal}
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\begin{document}$g(u)$\end{document} extend the prototypes f(u)=αus\documentclass[12pt]{minimal}
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\begin{document}$f(u)=\alpha u^{s}$\end{document} and g(u)=γur\documentclass[12pt]{minimal}
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\begin{document}$g(u)=\gamma u^{r}$\end{document} with α,γ>0\documentclass[12pt]{minimal}
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\begin{document}$\alpha ,\gamma >0$\end{document} and suitable s,r>0\documentclass[12pt]{minimal}
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\begin{document}$s,r>0$\end{document} for all u≥0\documentclass[12pt]{minimal}
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\begin{document}$u\ge 0$\end{document}. Our main result exhibits that there exists M∗>0\documentclass[12pt]{minimal}
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\begin{document}$M^{*}>0$\end{document} such that for all properly regular initial data, the studied model admits a unique classical solution which remains bounded if m2+s<m3+r\documentclass[12pt]{minimal}
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\begin{document}$m_{2}+s< m_{3}+r$\end{document} or m2+s=m3+r\documentclass[12pt]{minimal}
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\begin{document}$m_{2}+s=m_{3}+r$\end{document} and ξγχα>M∗\documentclass[12pt]{minimal}
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\begin{document}$\frac{\xi \gamma }{\chi \alpha }>M^{*}$\end{document}.
